Answer: True.
Step-by-step explanation:
The total number of possible digits in the number system : 10 (from 0 to 9)
If there are 6 randomly selected digits in an automobile license tag, and each digit must be one of the 10 integers (0-9), then the choices for each digit in license tag = 10
Fundamental counting principle , the total number of ways to make 6-digits license tag where the choices for each digit in license tag is 10 will be :
[tex]10\times10\times10\times10\times10\times10=10^6[/tex]
Hence, there are [tex]10^6[/tex] possible license tags.
Therefore , the given statement is correct.
Final answer:
The statement is True because for each of the 6 digits in a license tag, there are 10 possible integers, giving us a total of 10^6 possible combinations.
Explanation:
The question relates to probability and combinatorics, which are branches of mathematics that deal with the likelihood of certain outcomes and the combination of different elements, respectively. If an automobile license tag consists of 6 randomly selected digits and each digit must be one of the 10 integers (0-9), then we must consider every digit independently.
Since there are 10 options for each digit, and there are 6 digits, we calculate the total number of possible license tags by multiplying the number of options for each digit. This is done by raising the number of options (10) to the power of the number of digits (6), which gives us: 10^6. Therefore, the statement is True as there are indeed 10^6 possible license tags.
A farmer is building a rectangular pen along the side of a barn for animals. The barn will serve as one side of the pen. The farmer has 120 feet of fence to enclose an area of 1512 square feet and wants each side of the pen to be at least 20 feet long.Find the dimensions of the pen. How would I put this into my graphing calculator to solve?
The dimension of pen is 42 feet by 36 feet
Solution:
Let "x" be the width of pen
Let "y" be the length
Farmer wants each side of the pen to be at least 20 feet long
[tex]x\geq 20[/tex]
The farmer has 120 feet of fence to enclose an area of 1512 square feet
The amount of fencing is equal to the perimeter of fence which is 2 times the width plus only one length since the other side (length) is along the barn
2x + y = 120
Therefore,
y = 120 - 2x
The area of barn is given as 1512 square feet
The area of rectangle is given as:
[tex]Area = length \times width[/tex]
[tex]1512 = x \times y\\\\1512 = x \times (120-2x)\\\\1512 = 120x - 2x^2\\\\2x^2-120x + 1512 = 0[/tex]
Divide the entire equation by 2
[tex]x^2-60x + 756 = 0[/tex]
Factor the left side of equation
[tex]x^2-60x+756 = 0\\\\(x-18)(x-42) = 0[/tex]
Therefore, we get two values of "x"
x = 18 or x = 42
Since, [tex]x\geq 20[/tex]
Therefore, x = 42 is the solution
Thus, width = x = 42 feet
Length = y = 120 - 2x
y = 120 - 2(42)
y = 120 - 84
y = 36
Thus the dimension of pen is 42 feet by 36 feet
The area of a shape is the amount of space it occupies.
The dimension of the pen is 42 by 36 feet.
The perimeter is given as:
[tex]\mathbf{P = 120}[/tex]
Because one of the sides does not need fencing, the perimeter would be:
[tex]\mathbf{P = 2x + y}[/tex]
Make y the subject
[tex]\mathbf{y = P - 2x}[/tex]
Substitute 160 for P
[tex]\mathbf{y = 120 - 2x}[/tex]
The area of a pen is:
[tex]\mathbf{A = xy}[/tex]
Substitute [tex]\mathbf{y = 120 - 2x}[/tex]
[tex]\mathbf{A = x(120 -2x)}[/tex]
Substitute 1512 for Area
[tex]\mathbf{x(120 -2x) = 1512}[/tex]
Open brackets
[tex]\mathbf{120x -2x^2 = 1512}[/tex]
Rewrite as:
[tex]\mathbf{2x^2 -120x + 1512 = 0}[/tex]
Divide through by 2
[tex]\mathbf{x^2 -60x + 756 = 0}[/tex]
Expand
[tex]\mathbf{x^2 -18x - 42x + 756 = 0}[/tex]
Factorize
[tex]\mathbf{(x -18)(x - 42) = 0}[/tex]
Solve for x
[tex]\mathbf{x =18 \ or\ x = 42}[/tex]
The dimension must be at least 20.
So, we have:
[tex]\mathbf{x = 42}[/tex]
Recall that:
[tex]\mathbf{y = 120 - 2x}[/tex]
This gives:
[tex]\mathbf{y = 120 - 2 \times 42}[/tex]
[tex]\mathbf{y = 36}[/tex]
Hence, the dimension of the pen is 42 by 36 feet.
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A principal placed a total of 288 math books on 8 book charts. He placed an equal number of books on each cart. Each book is 6 pounds. How much pounds of books was placed on the last book cart.
Final answer:
The total weight of books on the last book cart is 216 pounds, found by dividing the total number of books by the number of carts and then multiplying by the weight per book.
Explanation:
The question involves dividing the total number of math books evenly among book carts and then determining the total weight of books on one cart. Here's a step-by-step solution:
First, divide the total number of books, which is 288, by the number of book carts, which is 8, to find the number of books per cart: 288 \/ 8 = 36 books per cart.
Next, since each book weighs 6 pounds, multiply the number of books per cart by the weight per book to get the total weight on one book cart: 36 books x 6 pounds/book = 216 pounds.
Therefore, the total weight of books on the last book cart is 216 pounds.
In two or more complete sentences, explain how you can use the two functions, C(m) and T(m) to determine how many miles, m, a car needs to be driven, during a one day car rental, in order for the total cost to be same at both car rental companies. In your final answer, include a solution for the number of miles, m.
C(m) charges a flat rate of $41 and an extra fee of $0.10 per mile driven.
T(m) charges a fee of $0.25 per mile driven.
To find the point at which the total cost of rental is the same for both companies, set the cost functions C(m) and T(m) equal to each other and solve for m. In this case, the car needs to be driven approximately 273 miles.
Explanation:To determine the number of miles, m, that a car needs to be driven in a day to equal the total cost at both rental companies, we need to find when C(m) is equal to T(m).
C(m) = $41 + $0.10m
T(m) = $0.25m
Setting these two equations equal to each other gives the equation $41 + $0.10m = $0.25m. Solve this equation to find the value of m.
Subtract $0.10m from both sides, leaving: $41 = $0.15m. Next, divide both sides by $0.15 to obtain the value of m, which is m = $41/$0.15 = 273.33. Therefore, the car needs to be driven approximately 273 miles for the rental cost to be the same at both companies.
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Please help! Just with Part B...
26 Five pounds of body fat is equivalent to 16 , 000 calories. Carol can burn 600 calories per hour bicycling and 400 calories per hour swimming.
a)
How many calories will Carol burn in x hours of cycling? How many calories will she burn in y hours of swimming?
b)
Write an equation in general form that relates the number of hours, x , of cycling and the number of hours, y , of swimming Carol needs to perform in order to lose 5 pounds.
Answer:
a) Number of calories burned in [tex]x[/tex] hours of cycling [tex]600x[/tex] and Number of calories burned in [tex]y[/tex] hours of Swimming is [tex]400y[/tex].
b)The equation in general form that relates number of hours of cycling and swimming carols needs to perform to loose 5 pounds is [tex]6x+4y=160[/tex].
Step-by-step explanation:
Given:
5 pounds of body fat = 16000 calories
Number of calories burn in 1 hour of cycling = 600
Number of calories burn in 1 hour of swimming = 400
Part a:
We need to find number of calories will Carol burn in [tex]x[/tex] hours of cycling and number of calories will Carol burn in [tex]y[/tex] hours of swimming.
Solution:
Now we know that;
1 hr of cycling = 600 calories burned
[tex]x[/tex] hr of cycling = Number of calories burned in [tex]x[/tex] hours of cycling.
By using Unitary method we get;
Number of calories burned in [tex]x[/tex] hours of cycling = [tex]600x[/tex]
Also we know that;
1 hr of swimming= 400 calories burned
[tex]y[/tex] hr of swimming = Number of calories burned in [tex]y[/tex] hours of swimming.
By using Unitary method we get;
Number of calories burned in [tex]y[/tex] hours of Swimming = [tex]400y[/tex]
Hence Number of calories burned in [tex]x[/tex] hours of cycling [tex]600x[/tex] and Number of calories burned in [tex]y[/tex] hours of Swimming is [tex]400y[/tex]
Part b:
We need to write equation in general form that relates number of hours of cycling and swimming carols needs to perform to loose 5 pounds.
Solution:
Now given;
5 pounds = 16000 calories.
So we can say that;
Total number of calories to be burn is equal to sum of Number of calories burned in [tex]x[/tex] hours of cycling and Number of calories burned in [tex]y[/tex] hours of Swimming.
framing in equation form we get;
[tex]600x+400y=16000[/tex]
Now dividing both side by 100 we get;
[tex]\frac{600x}{100}+\frac{400y}{100}=\frac{16000}{100}\\\\6x+4y=160[/tex]
Hence the equation in general form that relates number of hours of cycling and swimming carols needs to perform to loose 5 pounds is [tex]6x+4y=160[/tex].
Suppose the radius of the sphere is increasing at a constant rate of 0.3 centimeters per second. At the moment when the radius is 24 centimeters, the volume is increasing at a rate of?
Step-by-step explanation:
We have equation for volume of a sphere
[tex]V=\frac{4}{3}\pi r^3[/tex]
where r is the radius
Differentiating with respect to time,
[tex]\frac{dV}{dt}=\frac{d}{dt}\left (\frac{4}{3}\pi r^3 \right )\\\\\frac{dV}{dt}=\frac{4}{3}\pi \times 3r^2\times \frac{dr}{dt}\\\\\frac{dV}{dt}=4\pi r^2\times \frac{dr}{dt}[/tex]
Given that
Radius, r = 24 cm
[tex]\frac{dr}{dt}=0.3cm/s[/tex]
Substituting
[tex]\frac{dV}{dt}=4\pi r^2\times \frac{dr}{dt}\\\\\frac{dV}{dt}=4\pi \times 24^2\times 0.3\\\\\frac{dV}{dt}=2171.47cm^3/min[/tex]
At the moment when the radius is 24 centimeters, the volume is increasing at a rate of 2171.47 cm³/min.
i need help plz
Choose the word or phrase that best completes each sentence.
1. George Washington when he went to fight the French in the Ohio River Valley.
2. The British then sent a to attack the French Fort Duquesne.
3. The British in the first battle of the French Indian War.
Answer:
heres your answers
Step-by-step explanation:
George Washington was defeated when he went to fight the French in the Ohio River Valley.
The British then sent a trained army to attack the French Fort Duquesne.
The British suffered a defeat in the first battle of the French Indian War.
3. Use the Division Algorithm to establish the following: (a) The square of any integer is either of the form 3k or 3k + 1. (b) Thecubeofanyintegerhasoneoftheforms:9k,9k+1,or9k+8. (c) The fourth power of any integer is either of the form 5k or 5k + 1.
Explanation
Lets use congruence
For the square of an integer lets use congruence module 3
If the congruence module 3 is 0, then the congruence of the sqaure is 0² = 0If the congruence is 1, then the congruence of the square is 1² = 1If the congruence is 2, then the congruence of the square is 2² = 4 = 4-3 = 1 (4 and 1 are equal is Z₃)Thus, the square of an integer has the form 3k (the congruence is 0) or 3k+1 (the congruence is 1).
For the cube, lets use congruence module 9
if the congruence module 9 is 0, then the congruence of the cube is 0³ = 0if the congruence is 1, then the congruence of the cube is 1³ = 1if the congruence is 2, then the congruence of the cube is 2³ = 8if the congruence is 3, then the congruence of the cube is 3³ = 27 = 0if the congruence is k+3, for certain k, then the congruence of the cube is (k+3)³ = k³+9k²+27k+9 = k³. Hence the congruence of the cube is the same after addding 3 to a number. Thus, the congruence of a number with congruence 4,5,6,7 or 8 module 9 is obtained by computing the congruence module 9 of 1,2 or 3.With the argument given above, we obtain that the congruence module 9 of the cube of a number is always 0,1 ir 8, thus, the cube of an integer has the form 9k, 9k+1 or 9k+8.
As for the fourth power, we take congruence module 5:
If the congruence module 5 is 0, then the congruence of the fourth power is 0⁴ = 0if the congruence is 1, then the congruence of the foruth power is 1⁴ = 1If the congruence is 2, then the congruence of the fourth power is 2⁴ = 16 = 16-5*3 = 1If the congruence is 3, then the congruence of the fourth power is 3⁴ = 81 = 81-5*16 = 1If the congruence is 4, then the congruence of the fourth power is 4⁴ = 256 = 256-5*51 = 1In any case, the congruence of the fourth power module 5 is either 0 or 1, as a result, the fourth power has the form 5k or 5k+1.
Final answer:
Using the Division Algorithm, we show that the square of an integer is in the form of 3k or 3k+1, the cube is in the form of 9k, 9k+1, or 9k+8, and the fourth power is in the form of 5k or 5k+1.
Explanation:
The Division Algorithm states that if we divide any integer n by a positive integer d, there are unique integers q and r such that n = dq + r and 0 ≤ r < d. Using this and thinking about integers as being of the form 3q, 3q + 1, or 3q + 2 we can prove the following statements:
(a) The square of any integer:
Any integer can be written as 3k, 3k + 1, or 3k + 2. Squaring these forms:
(3k)² = 9k² = 3(3k²)(3k + 1)² = 9k² + 6k + 1 = 3(3k² + 2k) + 1(3k + 2)² = 9k² + 12k + 4 = 3(3k² + 4k + 1) + 1In each case, the square of an integer is either of the form 3k or 3k + 1.
(b) The cube of any integer:
Cubing the three forms:
(3k)³ = 27k³ = 9(3k³)(3k + 1)³ = 27k³ + 27k² + 9k + 1 = 9(3k³ + 3k² + k) + 1(3k + 2)³ = 27k³ + 54k² + 36k + 8 = 9(3k³ + 6k² + 4k) + 8The cube is of the form 9k, 9k + 1, or 9k + 8.
(c) The fourth power of any integer:
For 3k and 3k + 1, the fourth powers are:
(3k)⁴ = 81k⁴ = 5(16k⁴)(3k + 1)⁴ = 81k⁴ + 108k³ + 54k² + 12k + 1 = 5(16k⁴ + 21k³ + 10k² + 2k) + 1For 3k + 2, because 2⁴ = 16 is already a multiple of 5, (3k + 2)⁴ will also have the form 5m or 5m + 1. Therefore, the fourth power of an integer is either of the form 5k or 5k + 1.
At the zoo, three adult lions together eat 250 pounds of food a day. If two more adult lions joined the group and ate food at the same rate as the original three, how much food would the zoo need to provide all five lions each day?
416.67 pounds of food should be given all five lions each day
Solution:
Given that, three adult lions together eat 250 pounds of food a day
Thus, 3 adult lions = 250 pounds of food per day
Two more adult lions joined the group and ate food at the same rate as the original three
Now number of adult lions = 3 adult lions + 2 adult lions = 5 adult lions
Let "x" be the food ate by 5 adult lions
Thus we can say,
3 adult lions = 250 pounds of food per day
5 adult lions = "x" pounds of food per day
This forms a proportion and we can solve the sum by cross multiplying
[tex]\frac{3}{5} = \frac{250}{x}\\\\3 \times x = 250 \times 5\\\\3x = 1250\\\\x = 416.67[/tex]
Thus 416.67 pounds of food should be given all five lions each day
Carole needs 4pounds of nuts for her granola. She has 26ounces of walnuts and 28ounces of cashews. How many ounces of peanuts should she buy so she has 4pounds of nuts?
Answer:
10 oz
Step-by-step explanation:
there are 16 oz in a pound. 4 * 16 = 64 so she needs 64 oz in all. she has 26 oz of walnuts and 28 oz of cashews. add these together and subtract it from 64 to find how many oz of peanuts she needs.
26 + 28 = 54
64 - 54 = 10
she needs 10 oz of peanuts
Define relation R on the set of natural numbers as follows: xRy iff each each prime factor of x is a factor of y. Prove that X is a partial order.
Answer: This relation is reflexive, antisymmetric and transitive so it is a partial order relation.
Step-by-step explanation: A relation is called a partial order relation if and only if it is reflexive, antisymmetric and transitive. We will check these three characteristics for the given relation.
Reflexive: We need to have that for all [tex]x\in\mathbb{N}[/tex], [tex]xRx[/tex]. This is obviously true since each prime factor of [tex]x[/tex] is certainly a factor of [tex]x[/tex].
Antisymmetric: We need to show that for all [tex]x,y\in\mathbb{N}[/tex] if both [tex]xRy[/tex] and [tex]yRx[/tex] then it must be [tex]x=y[/tex]. To show this suppose that two, otherwise arbitrary, natural numbers [tex]x[/tex] and [tex]y[/tex] are taken such that [tex]xRy[/tex] and [tex]yRx[/tex]. The first of these two says that every prime factor of [tex]x[/tex] is a factor of [tex]y[/tex]. The second one says that every prime factor of [tex]y[/tex] is a factor of [tex]x[/tex]. This means that every prime factor of [tex]x[/tex] is also the prime factor of [tex]y[/tex] and that every prime factor of [tex]y[/tex] is the prime factor of [tex]x[/tex] i.e. that [tex]x[/tex] and [tex]y[/tex] have the same prime factors meaning that they have to be equal.
Transitive: The relation is called transitive if from [tex]xRy[/tex] and [tex]yRz[/tex] then it must also be [tex]xRz[/tex]. To see that this is true of the given relation take some natural numbers [tex]x,y[/tex] and [tex]z[/tex] such that [tex]xRy[/tex] and [tex]yRz[/tex]. The first condition means that each prime factor of [tex]x[/tex] is the factor of [tex]y[/tex] i.e. that all the prime factors of [tex]x[/tex] are contained among the prime factors of [tex]y[/tex]. The second condition means that each prime factor of [tex]y[/tex] is a factor of [tex]z[/tex] i.e. that all the prime factors of [tex]y[/tex] are contained among the prime factors of [tex]z[/tex]. So we have that all of the prime factors of [tex]x[/tex] are contained among the prime factors of [tex]y[/tex] and they themselves are contained among the prime factors of [tex]z[/tex]. This means that certainly all of the prime factors of [tex]x[/tex] are contained among the prime factors of [tex]z[/tex] meaning by the given definition of [tex]R[/tex] that [tex]xRz[/tex] which is what we needed to show.
The number of bocks has 9 in the ones place. The number in the hundreds place is one more then the tens place. Those two numbers equal 11. How many block are there?
There are 659 blocks
Solution:
The number in ones place is 9
Let's denote number in the tens place with x
The number in the hundreds place is one more then the tens place
Therefore,
The number in hundreds place is 1 + x
Those two numbers equal 11
Which means tens place number and hundreds place number sums up to 11
tens place + hundreds place = 11
x + 1 + x = 11
2x = 11 - 1
2x = 10
x = 5
Thus number in tens place = x = 5
Number in hundreds place = 1 + x = 1 + 5 = 6
The number is represented as:
Number = (Number in hundreds place)(number in tens place)(number in ones place)
Number = 659
Thus 659 blocks are there
A swimmer ascended in the pool 2/3 meters at a time. She did this 8 times to reach the surface of the pool. What is the distance that represents the swimmer's total ascension
Answer:
Step-by-step explanation:
So 8 times more than 2/3
The total ascension of the swimmer can be calculated by multiplying the distance ascended each time by the number of ascents, resulting in 5 and 1/3 meters.
Explanation:The total ascension of the swimmer can be calculated by multiplying the distance ascended each time by the number of ascents. In this case, the swimmer ascended 2/3 meters 8 times.
Total ascension = 2/3 meters * 8 ascents = 16/3 meters = 5 and 1/3 meters.
Therefore, the distance representing the swimmer's total ascension is 5 and 1/3 meters.
On Friday 537 people attended a play on Saturdays 812 people attended the same place about how many more people to know the play on Saturday then on Friday how did you estimate
Answer:
275 people more attended the play on Saturday then on Friday.
Step-by-step explanation:
Given:
number of people attended play on Friday = 537
Number of people attended play on Saturday = 812.
We need to find how many people more attended the play Saturday then on Friday.
Solution:
Now we can say that;
To find Number of people more attended the play on Saturday then on Friday can be calculated by subtracting the number of people attended play on Friday from Number of people attended play on Saturday.
framing in equation form we get;
Number of people more attended = [tex]812-537 = 275[/tex]
Hence 275 people more attended the play on Saturday then on Friday.
A random sample of n people selected from a large population will be asked whether they have read a novel in the past year. Let the random variable R represent the number of people from the sample who answer yes. The variance of random variable R is 6. Assume the responses are independent of each other. If the proportion of people from the population who read a novel in the past year is 0.40, which of the following is the best interpretation of random variable R ?
1. A binomial variable with 15 independent trials
2. A binomial variable with 25 independent trials
3. A variable that is not binomial with 25 independent trials
4. A binomial variable with 40 independent trials
5. A variable that is not binomial with 40 independent trials
Answer:
Option 2) A binomial variable with 25 independent trials
Step-by-step explanation:
We are given the following in the question:
Sample size = n
R: the number of people from the sample who answer yes to the question whether they have read a novel in the past year.
[tex]var(R) = 6[/tex]
[tex]p =0.40[/tex]
Then, the random variable R follows a binomial distribution:
There are n independent trials.Each trial have two results either they have read the book or they have not read the bookProbability of success for each trial is same.[tex]var(R) = npq\\6 = n(0.4)(1-0.4)\\\\n = \dfrac{6}{0.4\times 0.6}\\\\n = 25[/tex]
Thus, R is a binomial variable with 25 independent trials.
Answer:
A binomial variable with 25 independent trials
Harry's amount of money is 75% of Kayla's amount of money after harry earned $30 and Kayla earned 25% more of her money harry's amount of money is 80% of Kayla's money how much money did each have at the beginning
Answer:
Harry had $90 and Kayla had $120 at the beginning.
Step-by-step explanation:
Given:
Initial amount of Harry's money = 75% of Kayla's initial amount.
Harry earned $30 and Kayla earned 25% more.
Harry's final amount = 80% of Kayla's final amount.
Let the initial amount of Kayla be 'x'.
As per question:
Initial amount of Harry's money = 75% of Kayla's initial amount.
Initial amount of Harry's money = 75% of [tex]x[/tex]
Initial amount of Harry's money = [tex]0.75x[/tex]
Now, Harry earned $30 more. So, total amount of Harry is given as:
Final amount of Harry's money = [tex]0.75x+30[/tex]
Kayla also earned 25% more of her money. So, final amount of Kayla's money is given as:
Kayla's final amount = [tex]x+25\%\ of\ x=x+0.25x=1.25x[/tex]
Now, again as per question:
Harry's final amount = 80% of Kayla's final amount.
[tex]0.75x+30=0.80\times 1.25x[/tex]
[tex]0.75x+30=1x[/tex]
[tex]1x-0.75x=30[/tex]
[tex]0.25x=30[/tex]
[tex]x=\frac{30}{0.25}=\$120[/tex]
Therefore, initial money Kayla had = $120
Initial money Harry had = [tex]0.75x=0.75\times 120=\$90[/tex]
Hence, Harry had $90 and Kayla had $120 at the beginning.
In an opinion poll, 25% of a random sample of 200 people said that they were strongly opposed to having a state lottery. The standard error of the sample proportion is approximately_________.
Answer: 0.031 .
Step-by-step explanation:
The standard error of the sample proportion is given by :-
[tex]SE_p=\sqrt{\dfrac{p(1-p)}{n}}[/tex]
, where p= Sample proportion and n is the sample size.
As per given , we have
In an opinion poll, 25% of a random sample of 200 people said that they were strongly opposed to having a state lottery.
i.e. p= 0.25 and n= 200
Then , the standard error of the sample proportion [tex]=\sqrt{\dfrac{0.25(1-0.25)}{200}}[/tex]
[tex]=\sqrt{\dfrac{0.25\times0.75}{200}}=\sqrt{0.0009375}\\\\=0.0306186217848\approx0.031[/tex]
Hence, the standard error of the sample proportion is approximately 0.031 .
Final answer:
The standard error of the sample proportion is approximately 0.0306.
Explanation:
The standard error of a sample proportion can be calculated using the formula:
Standard Error = √((p)(1-p)/n)
where p is the proportion of the sample and n is the sample size. In this case, the proportion is 0.25 (since 25% of the sample is strongly opposed to having a state lottery) and the sample size is 200. Plugging these values into the formula:
Standard Error = √((0.25)(1-0.25)/200) = √(0.1875/200) = √(0.0009375) = 0.0306
So, the standard error of the sample proportion is approximately 0.0306.
Stephen & Richard share a lottery win of £2950 in the ratio 2 : 3. Stephen then shares his part between himself, his wife & their son in the ratio 3 : 5 : 2. How much more does his wife get over their son?
Answer:
Stephen's wife got £354 more than his son.
Step-by-step explanation:
Given:
Amount of Lottery = £2950
Now Given:
Stephen & Richard share a lottery amount in the ratio 2 : 3
Let the common factor between them be 'x'.
So we can say that;
[tex]2x+3x=2950\\\\5x = 2950[/tex]
Dividing both side by 5 we get;
[tex]\frac{5x}{5}=\frac{2950}{5}\\\\x = 590[/tex]
So we can say that;
Stephen share would be = [tex]2x =2\times 590 = \£1180[/tex]
Now Given:
Stephen then shares his part between himself, his wife & their son in the ratio 3 : 5 : 2.
Let the common factor between them be 'y'.
So we can say that;
[tex]3y+5y+2y=1180\\\\10y=1180[/tex]
Dividing both side by 10 we get;
[tex]\frac{10y}{10}=\frac{1180}{10}\\\\y=118[/tex]
So Stephen's wife share = [tex]5y = 5\times 118= \£590[/tex]
And Stephen's son share = [tex]2y=2\times118 =\£236[/tex]
Now we need to find how much more her wife got then her son.
To find how much more her wife got than her son we will subtract Stephen's son share from Stephen's wife share.
framing in equation form we get;
Amount more her wife got than her son = [tex]590-236 = \£354[/tex]
Hence Stephen's wife got £354 more than his son.
A man flies a kite at a height of 16 ft. The wind is carrying the kite horizontally from the man at a rate of 5 ft./s. How fast must he let out the string when the kite is flying on 34 ft. of string?
Answer:
4.41 feet per second.
Step-by-step explanation:
Please find the attachment.
We have been given that a man flies a kite at a height of 16 ft. The wind is carrying the kite horizontally from the man at a rate of 5 ft./s. We are asked to find how fast must he let out the string when the kite is flying on 34 ft. of string.
We will use Pythagoras theorem to solve for the length of side x as:
[tex]x^2+16^2=34^2[/tex]
[tex]x^2=34^2-16^2[/tex]
[tex]x^2=900\\\\x=30[/tex]
Now, we will use Pythagorean theorem to relate x and y because we know that the vertical side (16) is always constant.
[tex]x^2+16^2=y^2[/tex]
Let us find derivative of our equation with respect to time (t) using power rule and chain rule as:
[tex]2x\cdot \frac{dx}{dt}+0=2y\cdot \frac{dy}{dt}[/tex]
We have been given that [tex]\frac{dx}{dt}=5[/tex] , [tex]y=34[/tex] and [tex]x=30[/tex].
[tex]2(30)\cdot 5=2(34)\cdot \frac{dy}{dt}[/tex]
[tex]300=68\cdot \frac{dy}{dt}[/tex]
[tex]\frac{dy}{dt}=\frac{300}{68}[/tex]
[tex]\frac{dy}{dt}=4.4117647058823529[/tex]
[tex]\frac{dy}{dt}\approx 4.41[/tex]
Therefore, the man must let out the string at a rate of 4.41 feet per second.
Significant Figures: The last page of a book is numbered 764. The book is 3.0 cm thick, not including its covers. What is the average thickness (in centimeters) of a page in the book, rounded to the proper number of significant figures?
Answer:
0.0079 cm.
Step-by-step explanation:
We have been given that the last page of a book is numbered 764. The book is 3.0 cm thick, not including its covers.
We know that each page is marked on both sides, so we will find total number of pages by dividing 764 by 2 as:
[tex]\text{Total number of pages}=\frac{764}{2}[/tex]
[tex]\text{Total number of pages}=382[/tex]
To find the average thickness of each page, we will divide thickness of book by total number of pages as:
[tex]\text{Average thickness of each page}=\frac{3.0}{382}[/tex]
[tex]\text{Average thickness of each page}=0.0078534031413613[/tex]
We can see that there are 3 significant figures in 764 and 2 significant digits in 3.0.
We know that the result of a multiplication or division is rounded to the number of significant figures equal to the smallest number of significant figures among the numbers being multiplied/divided. So we need to round our answer to 2 significant digits.
[tex]\text{Average thickness of each page}\approx 0.0079[/tex]
Therefore, the average thickness of each page is approximately 0.0079 cm.
Need help ASAP!! will mark brainliest for fast response.
(attached screenshot)
Answer:
hi
Step-by-step explanation:
12 a 2 in box # 1
4-6a for box # 2
2a*2+8a box # 3
12a*2- 8 a box # 4
have a good day hope this helps
Answer:
The answer to your question is below
Step-by-step explanation:
Remember that the greatest common factor of a set of numbers is the greatest factor common to all the numbers.
Greatest common factor Factor Expression
2a² + 8a 2a(a + 4) 2a
12a² - 8a 4a(3a - 2) 4a
4a + 8 4(a + 2) 4
4 - 6a 2(2 - 3a) 2
A football field is sloped from the center toward the sides for drainage. The height h, in feet, of the field x feetfrom, the side, is given by h=-0. 00025x2 + 0 04x. Find the height of the field a distance of 35 feet from the side The height of the field is □ feet. Round to the nearest tenth as needed.
To find the height of the football field at a distance of 35 feet from the side, you plug x=35 into the equation h=-0.00025x^2 + 0.04x, resulting in a height of 1.1 feet after rounding to the nearest tenth.
Explanation:To find the height of the football field a distance of 35 feet from the side, we will substitute the value of x into the given quadratic equation h = -0.00025x2 + 0.04x. We can follow these steps:
First, plug in the value of x, which is 35, into the equation: h = -0.00025(35)2 + 0.04(35).Next, calculate the square of 35, which is 1225, and then multiply by -0.00025 to get -0.30625.Then, multiply 0.04 by 35 to get 1.4.Now, add the two results to get the height: h = -0.30625 + 1.4.After performing the addition, the height is h = 1.09375 feet.Lastly, round the result to the nearest tenth: the height of the field is 1.1 feet at a distance of 35 feet from the side.The process of using variables to represent unknown quantities and then finding relationships that involve these variables is referred to as ____________.
Answer:
Mathematical modeling
Step-by-step explanation:
Mathematical modeling is defined as translating the problems from an application area using the mathematical formulas.
The numerical analysis and theoretical analysis gives an insight and answers or guidance which is useful for the originating application.
It provides with the precision and direction for the solution of the problem.
In this problem, y = c₁eˣ + c₂e⁻ˣ is a two-parameter family of solutions of the second-order DE y'' − y = 0. Find a solution of the second-order IV P consisting of this differential equation and the given initial conditions. y(-1) = 4, y'(-1) = -4.
Answer:
[tex]y=4e^{-(x+1)}[/tex] will be the solutions.
Step-by-step explanation:
The given equation is [tex]y=C_{1}e^{x}+C_{2}e^{-x}[/tex]
Therefore, for x = -1
[tex]4=C_{1}e^{-1}+C_{2}e^{1}[/tex] ------(1)
Now y'(-1) = -4
y'(x) = [tex]C_{1}e^{x}-C_{2}e^{-x}[/tex] = -4
[tex]C_{1}e^{-1}-C_{2}e^{1}[/tex] = -4 -----(2)
By adding equation (1) and (2)
[tex]2C_{1}e^{-1}=0[/tex]
[tex]C_{1}=0[/tex]
From equation (1),
[tex]4=0+C_{2}e^{1}[/tex]
[tex]C_{2}=4e^{-1}[/tex]
By placing the values in the parent equation
y = [tex]4e^{-1}\times e^{-x}[/tex]
y = [tex]4e^{-(x+1)}[/tex]
You have a coupon worth $18 off the purchase of a scientific calculator.The calculator is also being offered at a discount of 15% off. You can only use either the coupon or the discount, but not both.Write a linear equation to represent using the coupon.Use x for the cost of the calculator And y for the cost after the coupon.
Answer:
The linear equation representing cost after using coupon is [tex]y=x-18[/tex].
Step-by-step explanation:
Given:
Value of coupon = $18
Let the cost of the calculator be 'x'.
And the Cost after redeeming the coupon be 'y'.
We need to write a linear equation to represent using the coupon.
Solution:
Now we can say that;
Cost after redeeming the coupon will be equal to cost of the calculator minus Value of coupon.
framing in equation form we get;
[tex]y=x-18[/tex]
Hence The linear equation representing cost after using coupon is [tex]y=x-18[/tex].
A company produces three combinations of mixed vegetables that sell in 1-kg packages. Italian style combines 0.4 kg of zucchini, 0.3 kg of broccoli, and 0.3 kg of carrots. French style combines 0.5 kg of broccoli and 0.5 kg of carrots. Oriental style combines 0.2 kg of zucchini, 0.3 kg of broccoli, and 0.5 kg of carrots. The company has a stock of 18 comma 200 kg of zucchini, 28 comma 100 kg of broccoli, and 38 comma 700 kg of carrots. How many packages of each style should it prepare to use up existing supplies?
Answer:
Italian style = 19,000 packages
French style = 13,000 packages
Oriental style = 53,000 packages
Step-by-step explanation:
let the number of packages of Italian style = x
let the number of packages of French style = y
let the number of packages of Oriental style = z
See the attached table which summarize the problem
Using the table we can get the following system of equations:
0.4x + 0 * y + 0.2z = 18,200
0.3x + 0.5y + 0.3z = 28,100
0.3x + 0.5y + 0.5z = 38,700
Solving the 3 equations together to find x , y and z
Using the calculator
x = 19,000
y = 13,000
z = 53,000
The relationship can be modeled by the quadratic equation p = − 100s^2 + 2400s − 8000, where p represents the profit and s represents the selling price. Which selling price will maximize profits? A.$12 B.$20 C.$8 D.$4
Answer:
A. $12
Step-by-step explanation:
For a quadratic of the form ax²+bx+c, the axis of symmetry is given by ...
x=-b/(2a)
For your quadratic function, the axis of symmetry is ...
s = -(2400)/(2(-100)) = 12
The function has its extreme value on the axis of symmetry. Here, the leading coefficient is negative, so the parabola opens downward. That means the extreme value is a maximum.
Profit will be a maximum at a selling price of $12.
Peter wants to buy a coat that costs $87 at full price. The coat is now on sale for 40% off. Part A What is the amount in dollars he will save on the coat?
Answer:
34.8
Step-by-step explanation:
1. Convert percent to decimal
40.00/100=.4
2.Multiply decimal by subtotal
87*.4=34.8
Peter will save $34.80 on the coat.
What is Percentage?Percentage is a way of expressing a proportion or a fraction as a number out of 100. It is often denoted by the symbol "%".
If the coat is on sale for 40% off, Peter will only need to pay 60% of the original price.
60% of $87 can be calculated as follows:
= 60/100 x $87
= $52.20
Therefore, the amount that Peter will save on the coat is:
= $87 - $52.20
= $34.80
So, Peter will save $34.80 on the coat.
Learn more about Percentage here:
https://brainly.com/question/29306119
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The American Red Cross says that about 45% of the U.S. population has Type O blood, 40% Type A, 11% Type B, and the rest Type AB.
a) One person volunteers to give blood. What is the probability they don't have Type O blood?
b) Among four potential donors (chosen at random), what is the probability that all of them have Type A blood?
c) Among four potential donors (chosen at random), what is the probability that none of them have Type O blood?
d) Among four potential donors (chosen at random), what is the probability that at least one person has Type O blood?
Answer:
a) The probability is %55
b) The probability is %0.15
c) The probability is %9.15
d) The probability is %91.85
Step-by-step explanation:
a) We need to basically subtract probability of 0 type blood from 1:
P=1-0.45=0.55
b) The probability of one person that is having A blood type is %11. Then probability of four persons that are having A blood type will be:
(0.11)^4=0.00015
c) We need to approach to this question same as B. Probability of having not 0 type of blood is %55. Then probability of four persons that are not having 0 type of blood will be:
(0.55)^4=0.0915
d) To find the probability we can simply subtract probability of four persons that are having 0 type blood from 1:
1-0.0915=0.9185
Which would be the best trend line for the given data set?
A. y=-3/2x+8
B. y=3/2x+5
C. y=2/3x+8
D. y=-2/3x+5
Chase purchased a sweatshirt from the clearance rack. The price of the sweatshirt after the discount is represented by the expression 0.75x,where x represents the original price of the sweatshirt
Question is Incomplete; Complete question is given below;
Chase purchased a sweatshirt from the clearance rack. The price of the sweatshirt after the discount is represented by the expression 0.75x, where x represents the original price of the sweatshirt.
Which expression also represents the discounted price of the sweatshirt?
A (0.75 − 0.25)x
B (0.75 + 0.25)x
C x – 0.25x
D x + 0.25x
Answer:
C . [tex]x-0.25x[/tex]
Step-by-step explanation:
Given:
Expression representing price of the sweatshirt after the discount = [tex]0.75x[/tex]
[tex]x[/tex] ⇒ original price of sweatshirt
We need to find the expression which also represents the discounted price of the sweatshirt.
Solution:
Form the given expression we can see that after discount we are paying only 75% of the original amount of sweatshirt.
So we can say that;
The discount price was 25% of the original price i.e [tex]0.25x[/tex]
So now we can say that;
Price after discount is equal to difference of original price and discounted price.
framing in equation form we get;
Price after discount = [tex]x-0.25x[/tex]
Hence the equivalent expression for the given discounted price of sweatshirt is [tex]x-0.25x[/tex].
Answer:
Step-by-step explanation: