Answer:
[tex]g(n)=g(n/2)+n/2[/tex]
Step-by-step explanation:
[tex](2k+1)!=1*3*5*7*...*(2k-1)*(2k+1)*2*4*6*8*...*(2k-2)*(2k)[/tex]
Notice that the odd factors of above equality don't contribute to the largest power of 2 that divides (2k+1)!
Therefore, we can conclude that [tex]g(2n+1)=g(2n)[/tex].
[tex](2k)!=1*3*5*7*...*(2k-3)*(2k-1)*2*4*6*8*...*(2k-2)*(2k)=\\= (1*3*5*...*(2k-3)*(2k-1))*((2*1)*(2*2)*(2*3)*...*(2*(k-1))*(2*k))=\\= (1*3*5*...*(2k-3)*(2k-1))*2^k*k![/tex]
Notice that the odd factors of above equality don't contribute to the largest power of 2 that divides (2k)!
Hence, [tex]g(2k)-g(k)=k[/tex]
So for [tex]k = n/2[/tex],
[tex]g(n)-g(n/2)=n/2\\g(n)=g(n/2)+n/2[/tex]
Let $f(x) = x^2$ and $g(x) = \sqrt{x}$. Find the area bounded by $f(x)$ and $g(x).$
Answer:
[tex]\large\boxed{1\dfrac{1}{3}\ u^2}[/tex]
Step-by-step explanation:
Let's sketch graphs of functions f(x) and g(x) on one coordinate system (attachment).
Let's calculate the common points:
[tex]x^2=\sqrt{x}\qquad\text{square of both sides}\\\\(x^2)^2=\left(\sqrt{x}\right)^2\\\\x^4=x\qquad\text{subtract}\ x\ \text{from both sides}\\\\x^4-x=0\qquad\text{distribute}\\\\x(x^3-1)=0\iff x=0\ \vee\ x^3-1=0\\\\x^3-1=0\qquad\text{add 1 to both sides}\\\\x^3=1\to x=\sqrt[3]1\to x=1[/tex]
The area to be calculated is the area in the interval [0, 1] bounded by the graph g(x) and the axis x minus the area bounded by the graph f(x) and the axis x.
We have integrals:
[tex]\int\limits_{0}^1(\sqrt{x})dx-\int\limits_{0}^1(x^2)dx=(*)\\\\\int(\sqrt{x})dx=\int\left(x^\frac{1}{2}\right)dx=\dfrac{2}{3}x^\frac{3}{2}=\dfrac{2x\sqrt{x}}{3}\\\\\int(x^2)dx=\dfrac{1}{3}x^3\\\\(*)=\left(\dfrac{2x\sqrt{x}}{2}\right]^1_0-\left(\dfrac{1}{3}x^3\right]^1_0=\dfrac{2(1)\sqrt{1}}{2}-\dfrac{2(0)\sqrt{0}}{2}-\left(\dfrac{1}{3}(1)^3-\dfrac{1}{3}(0)^3\right)\\\\=\dfrac{2(1)(1)}{2}-\dfrac{2(0)(0)}{2}-\dfrac{1}{3}(1)}+\dfrac{1}{3}(0)=2-0-\dfrac{1}{3}+0=1\dfrac{1}{3}[/tex]
The area bounded by the functions f(x) and g(x) in graph below.
The given function are f(x)=x² and g(x)=√x.
What is the function?Functions are the fundamental part of the calculus in mathematics. The functions are the special types of relations. A function in math is visualized as a rule, which gives a unique output for every input x.
To find the area between two curves defined by functions, integrate the difference of the functions. If the graphs of the functions cross, or if the region is complex, use the absolute value of the difference of the functions.
Area bounded = |x²-√x|
Find the domain by finding where the expression is defined.
Interval Notation:
[0,∞)
Set-Builder Notation:{x|x≥0}
Therefore, the area bounded by the functions f(x) and g(x) in graph below.
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If x is an integer greater than 1, is x equal to the 12th power of an integer ?(1) x is equal to the 3rd Power of an integer(2) x is equal to the 4th Power of an integer.
Answer:
The statement is true only when both (1) and (2) are valid. If only one of (1) and (2) is valid, them the statement is not true.
Step-by-step explanation:
(1) alone is not sufficient, 27 is 3³ but is not a 12th power
(2) alone is not sufficient either, 81 is 3³ but it is not a 12th power
If both (1) and (2) are valid, then for each prime p that divides x, p should divide y and z, with y³ = x and z⁴=x.
Lets suppose that k is the highest power of p that divides y and m is the highest power that divides z, then (p^k)³ = (p^m)⁴. Therefore
p^3k = p^4m
This means that the power of p that appears on x is a multiple of both 3 and 4. Since those numbers are coprime, then that power is a multiple of 12.
This ensures that every prime dividing x has at least a power of 12 in the prime factirization, hence x is a 12th power.
A real estate developer builds a small number of model homes, one each in a different style. During an open house event, there is a lot of interest, but prospective buyers almost exclusively prefer the "Model C" style. At the conclusion of the open house event, the developer begins ramping up production quickly, doubling the number of "Model C" houses completed each month. While she does not tear them down, she decides not to build more of the other 5five styles of houses. In the month when the first residents begin to move in, two hundred sixty one total houses have been built, including the original model homes.
I need to write an equation to model the number of houses built over time. It has to be in y=a * b^x-k + h form, but there doesn't have to be all of them.
Edit : screw all of you guys for not answering
Answer:
y = 2^x +5
Step-by-step explanation:
The number of new houses doubles each month, so the base (b) is that multiplier, 2. Then x is the number of months since the construction started (presumably at the end of the open house). If you want x to be something else, then h needs to be chosen so that (x -k) is the number of months the doubling has been taking place.
The value h represents the initial build of 5 houses. This is confirmed by the number 261 = 256 + 5 = 2^8 +5.
So, your equation for the number of houses built over time is ...
y = 2^x +5
Determine the shape and bond angle of these oxynitrogen ions.
Linear 180°
Trigonal planar 120°
Tetrahedral 109.5°
Bent 120°
Bent 109°
Pyramidal109°
The shape and bond angle of oxy-nitrogen ions is determined by the number of bonds and lone pairs around the central atom, with examples being trigonal pyramidal for nitrogen with three bonds and one lone pair, tetrahedral for carbon with four bonds, and bent for oxygen with two bonds and two lone pairs.
The oxynitrogen ions have specific shapes and bond angles that correlate with their electron-pair geometries. For example, a nitrogen atom with three bonds and one lone pair will have a trigonal pyramidal shape with bond angles close to 109°. Carbon atoms in different environments also exhibit varied shapes: a carbon atom in CH₂ with four bonds and no lone pairs has a tetrahedral shape with bond angles of 109.5°, while a carbon atom in CO₂ with two double bonds (treated as equivalent to three bonds) and no lone pairs forms a trigonal planar shape.
An oxygen atom in OH with two bonds and two lone pairs has a bent or angular shape with bond angles close to 109°. It's essential to note that electron pair repulsion and hybridization of orbitals play a crucial role in determining these geometric shapes and the bond angles.
Based on the options provided and the common shapes of oxynitrogen ions, the correct answer would be:
- Trigonal planar, 120°
- Tetrahedral, 109.5°
- Bent, 120°
- Pyramidal, 109°
These shapes and bond angles are commonly observed in oxynitrogen ions depending on the arrangement of atoms and lone pairs around the central nitrogen atom.
The shape and bond angle of oxynitrogen ions depend on the arrangement of atoms and lone pairs around the central nitrogen atom. Let's go through each option:
1. Linear, 180°: This shape occurs when there are no lone pairs on the central nitrogen atom, and there are two bonded atoms around it. The bond angle is indeed 180°.
2. Trigonal planar, 120°: This shape occurs when there is one lone pair and three bonded atoms around the central nitrogen atom. The bond angle is indeed 120°.
3. Tetrahedral, 109.5°: This shape occurs when there are four bonded atoms around the central nitrogen atom and no lone pairs. The bond angle is indeed 109.5°.
4. Bent, 120°: This shape occurs when there is one lone pair and two bonded atoms around the central nitrogen atom. The bond angle is close to 120° but may deviate slightly due to lone pair-bond pair repulsions.
5. Bent, 109°: This shape is less common for oxynitrogen ions but can occur if there is a lone pair and two bonded atoms around the central nitrogen atom. The bond angle would be closer to 109° due to lone pair-bond pair repulsions.
6. Pyramidal, 109°: This shape occurs when there is one lone pair and three bonded atoms around the central nitrogen atom. The bond angle is indeed close to 109°.
Based on the options provided and the common shapes of oxynitrogen ions, the correct answer would be:
- Trigonal planar, 120°
- Tetrahedral, 109.5°
- Bent, 120°
- Pyramidal, 109°
These shapes and bond angles are commonly observed in oxynitrogen ions depending on the arrangement of atoms and lone pairs around the central nitrogen atom.
What is the equation of the following graph?
1. F(x) = tan x-2
2. F(x) = tan x + 2
3. F(x) = 2tan x
4. F(x) = tan x -3
Answer:
The required function is F(x) = tanx - 2.
Step-by-step explanation:
We all know the graph of tanx ,
The graph of tanx passes through the origin and repeats in the interval of π.
Now , after looking into the graph we can clearly say that it is the shifted graph of tanx , Which is shifted in the -y direction .
Thus ,
It is clear that F(x) must be like F(x) = tanx - c , where c is a positive constant .
Also, At x = 0 , Value of the function is coming out to be -2 ,
Putting x = 0 in F(x) = tanx - c , we get , F(x) = -c .
Thus c = 2.
So, the required function is F(x) = tanx - 2.
Malcolm is trying a very low-carbohydrate diet. He would like to keep the amount of carbs consumed in grams between the levels shown in the following compound inequality:
50 < 2x + 10 and 2x + 10 < 110
Solve for x in this inequality, and explain what the answer represents.
The solution to the compound inequality is 20 < x < 50. Malcolm should consume more than 20 but less than 50 grams of carbohydrates to adhere to his low-carbohydrate diet.
Explanation:To solve the compound inequality 50 < 2x + 10 and 2x + 10 < 110 for Malcolm's diet, we'll deal with each part of the inequality separately and then combine the results to find the range of values for x that satisfies both conditions.
For the first part, subtract 10 from both sides: 50 - 10 < 2x, which simplifies to 40 < 2x. Then, divide both sides by 2 to isolate x: 20 < x.For the second part, subtract 10 from both sides: 2x + 10 - 10 < 110 - 10, simplifying to 2x < 100. Dividing both sides by 2 yields: x < 50.Combining these inequalities gives us 20 < x < 50. This means that Malcolm should consume more than 20 but less than 50 grams of carbohydrates to stay within his targeted dietary range on his low-carbohydrate diet.
Omar recorded the number of hours he worked each week for a year. Below is a random sample that he took from his data. 13, 17, 9, 21 What is the standard deviation for the data? Standard deviation: s = StartRoot StartFraction (x 1 minus x overbar) squared + (x 2 minus x overbar) squared + ellipsis + (x n minus x overbar) squared Over n minus 1 EndFraction EndRoot. 0 4.4 5.2 26.7
Final answer:
The standard deviation for the given data is approximately 4.47.
Explanation:
The standard deviation is a measure of how spread out the data is from the mean. To calculate the standard deviation, we follow a formula that involves finding the difference between each data point and the mean, squaring those differences, and taking the average of the squared differences. Finally, we take the square root of the average to get the standard deviation.
For the given data, which is 13, 17, 9, and 21, we first find the mean (average) which is (13 + 17 + 9 + 21) / 4 = 60 / 4 = 15. Next, we calculate the squared differences from the mean for each data point:
(13 - 15)^2 = 4
(17 - 15)^2 = 4
(9 - 15)^2 = 36
(21 - 15)^2 = 36
Then, we find the average of the squared differences:
(4 + 4 + 36 + 36) / 4 = 80 / 4 = 20
Finally, we take the square root of the average:
sqrt(20) ≈ 4.47
So, the standard deviation for the given data is approximately 4.47.
Xander collected four times as many stamps as his cousin.If xander collected 60 stamps,then how many did his cousin collect What is the variable and the Equation
Answer:the number of stamps that his cousin collected is 20. The variable is x which represents number of stamps. The equation is
4x = 60
Step-by-step explanation:
Let x represent the number of stamps that his cousin collected.
Xander collected four times as many stamps as his cousin. This means that the number of stamps that Xander collected is 4x
If xander collected 60 stamps, then it means that
4x = 60
x = 60/4 = 20 stamps
Final answer:
The number of stamps Xander's cousin collected is determined by dividing the number of stamps Xander collected, which is 60, by 4. Therefore, his cousin collected 15 stamps.
Explanation:
If Xander collected four times as many stamps as his cousin, and Xander collected 60 stamps, we can determine how many stamps his cousin collected by setting up an equation. Let's define c as the number of stamps Xander's cousin collected. According to the information provided, Xander collected four times this amount, so we have the equation 4c = 60. To find the value of c, we divide both sides of the equation by 4:
c = 60 / 4
c = 15
Therefore, Xander's cousin collected 15 stamps.
Last year the girls' basketball team had 8 fifth-grade students and 7 sixth-grade students. What was the ratio of sixth-grade students to fifth-grade students on the team?
The ratio of sixth-grade students to fifth-grade students is 7/8 or 7:8
Step-by-step explanation:
Let us define ratio first.
"Ratio is the quantitative relationship between two quantities which tells that one quantity is how many times of another quantity"
Given
Number of fifth grade students = 8
Number of sixth grade students = 7
We have to find the ratio of sixth grade students to fifth grade students
So,
[tex]r = \frac{sixth-grade\ students}{fifth-grade\ students}\\r = \frac{7}{8}\\r = 7:8[/tex]
Hence,
The ratio of sixth-grade students to fifth-grade students is 7/8 or 7:8
Keywords: ratio, fraction
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An investor invested a total of $2,300 in two mutual funds. One fund earned a 8% profit while the other earned a 2% profit. If the investors total profit was $82, how much was invested in each mutual fund
Answer: the amount invested on the first mutual fund is $600 and the the amount invested on the second mutual fund is $1700
Step-by-step explanation:
Let x represent the amount of money that the investor invested in one mutual fund.
Let y represent the amount of money that the investor invested in the second mutual fund.
An investor invested a total of $2,300 in two mutual funds. This means that
x + y = 2300
Considering the first mutual fund, it earned a 8% profit. This means that amount earned is
8/100 × x = 0.08x
Considering the second mutual fund, it earned a 2% profit. This means that amount earned is
2/100 × y = 0.02y
If the investors total profit was $82, it means that
0.08x + 0.02y = 82 - - - - - - - - 1
Substituting x = 2300 - y into equation 1, it becomes
0.08(2300 - y) + 0.02y = 82
184 - 0.08y + 0.02y = 82
- 0.08y + 0.02y = 82 - 184
- 0.06y = - 102
y = - 102/- 0.06 = 1700
x = 2300 - y = 2300 - 1700
x = 600
What error did the student make in their work?
A) There is a calculation error (added, subtracted, multiplied or divided incorrectly).
B) BD is the midsegment and should be multiplied by 2, not AE.
C) BD is the midsegment and should be multiplied by 1/2, not AE.
D) AE is the midsegment and should be multiplied by 1/3.
Answer:
B
Step-by-step explanation:
just swap around which one you multiply by 2 as they have multiplied the larger one by 2 instead of the smaller one.
Answer:
B.
Step-by-step explanation:
The error is on the first line:
4x + 20 = 2(3x + 5) is correct
- because AE = 2BD.
The price of a certain painting increased by 20% during the first year and decreased by 15% during the second year. The price of the painting at the end of the 2-year period was what percent of the original price?
A. 102%B. 105%C. 120%D. 135%E. 140%
Answer:
A.102%
Step-by-step explanation:
Let cost price of painting=$100
In first year price increased 20%
Then , the price=[tex]100+100(0.20)=[/tex]$120
In second year
Price decreased 15%
Then , the price of painting=[tex]120-120(0.15)[/tex]
The price of painting=$102
Percent =[tex]\frac{final\;price}{Initial\;price}\times 100[/tex]
By using this formula
Then, we get
Percent of the original price=[tex]\frac{102}{100}\times 100[/tex]
Percent of the original price=102%
Option A is true.
Use matrices to determine the vertices of the reflected figure. Then graph the image and the pre-image on the same coordinate grid.
Triangle XYZ with vertices X(2, -9), Y(-6, 0) and Z(-5, -5), reflected over the y-axis.
A.
B.
C.
D.
(pictures below)
Answer:
see below for the matrix multiplicationgraph CStep-by-step explanation:
Reflection across the y-axis is a left-right reflection that changes only the sign of the x-coordinate. The transformation matrix (as for any rotation and/or reflection) can be 2-dimensional, and can left-multiply the coordinate matrix that has coordinate pairs as column vectors. The transformed coordinates show up as column vectors in the result matrix.
The only graphs showing left-right reflections are those of choices C and D. Only choice C has the points properly plotted on the graph.
The correct answer is graph C, or the third option.
Just got it right on edge 2020, hope this helps!! :)
find the radius of the oblique cylinder (PLEASE HELP EXTRA POINTS)
Answer:
The radius of the oblique cylinder is 4 cm.
Step-by-step explanation:
Given:
The volume of the oblique cylinder = [tex]192 \pi cm^3[/tex]
Height of the cylinder = 12 cm
To Find:
Radius of the oblique cylinder = ?
Solution:
we know that the volume of the oblique cylinder
=>[tex]\text{base area of the cylinder} \times \text{height of the cylinder}[/tex]------------------------------(1)
where base area of the cylinder is the area of the circle
so area of the circle = base area
[tex]\pi r^2[/tex] = base area---------------(2)
Substituting (2) in (1)
[tex]192 \pi = \pi r^2 \times height[/tex]
[tex]192 \pi = \pi \times r^2 \times height[/tex]
[tex]192 \pi = \pi \times r^2 \times 12[/tex]
[tex]\frac{192 \pi}{ \pi \times 12}= r^2[/tex]
[tex]\frac{192}{12}= r^2[/tex]
[tex]16= r^2[/tex]
[tex] r^2 =16[/tex]
[tex] r =\sqrt{16}[/tex]
r=4
How do you do this question?
Answer:
7
Step-by-step explanation:
To find the maximum possible value of f(2), assume f(x) is a line with the greatest possible slope. In this case, 5.
f(x) = 5x − 3
f(2) = 5(2) − 3
f(2) = 7
Each of the integers $1,$ $2,$ $3,$ $\dots,$ $16$ is written on a separate slip of paper and these slips are placed in a pile. Jillian will randomly draw slips from the pile without replacement and will continue drawing until two of the numbers she has drawn from the pile have a product that is a perfect square. What is the maximum number of slips that Jillian can draw without obtaining a product that is a perfect square?
Answer:
11 slips
Step-by-step explanation:
A perfect square is a positive integer that is the square of another integer. For example, 25 is a perfect square of 5. See the calculation below
.
[tex]5^{2} = 5*5\\5^{2} = 25[/tex]
There is a rule that should be kept in mind
1. When two perfect squares are multiplied by each other (e.g. 4 * 9), the result is a perfect square ([tex]36 = 6^{2}[/tex])
Let's identify the combination of numbers that result in perfect square, when multiplied with each other. These combinations are as follows,
• 1*4, 1*9, 1*16
• 2*8
• 3*12
• 4*9, 4*16
• 9*16
From the list of numbers 1, 4, 9 and 16 are already perfect square e.g. [tex]2^{2} = 4, 3^{2} = 9[/tex]. If they are multiplied by each other, the result will also be a perfect square. Let’s assume that our first number is 1. Now we can't have any of the three numbers (except for 1), mentioned above. This rule out these three numbers.
Next, from 2, 8, 3 and 12 we can only draw two numbers. e.g. if we take 2, we can’t take 8 as it will give a perfect square. Same goes with 3 and 12. Hence from these four numbers we can discard two of them.
We discarded three numbers initially and two now. Therefore, out of 16 slipds we can draw a maximum of 11 slips without obtaining a product that is a perfect square.
A regular hexagon is inscribed in a circle, the diameter of the circle is d; what is the circumference of the hexagon ?
Answer:
3d
Step-by-step explanation:
A regular hexagon has six equivalent triangles, each triangle having one side of that of the hexagon. if the diameter of the circle in which the hexagon is inscribed is d , then its radius will be
[tex]r=\frac{d}{2}[/tex]
this radius also happens to be one of the side of the equivalent triangle whose one side is also the side of hexagon. since length of one side of hexagon is
[tex]\frac{d}{2}[/tex]
the circumference of the hexagon will be [tex]6 (\frac{d}{2} )\\\\thus \\circumference \\of \\hexagon = 3d[/tex]
Jolene entered an 18-month investment contract that guarantees to pay 2 percent interest at the end of 6 months, another 3 percent interest at the end of 12 months, and 4 percent interest at the end of the 18 month contract. If each interest payment is reinvested in the contract, and Jolene invested $10,000 initially, what will be the total amount of interest paid during the 18-month contract?
A. $506.00
B. $726.24
C. $900.00
D. $920.24
E. $926.24
Answer:
E. $926.24
Step-by-step explanation:
The total amount of interest paid is shown below:
1. For 6 months, the interest would be
= Invested amount × interest rate
= $10,000 × 2%
= $200
2. For 12 months, the interest would be
= (Invested amount + interest paid for 6 months) × interest rate
= ($10,000 + $200) × 3%
= $10,200 × 3%
= $306
3. For 18 months, the interest would be
= (Invested amount + interest paid for 6 months + interest paid for 12 months ) × interest rate
= ($10,000 + $200 + $306) × 4%
= $420
Now the total interest would be
= $200 + $306 + $420.24
= $926.24
The legs of a right triangle are in the ratio of 3 to 1. If the length of the hypotenuse of the triangle is 40√40, then the perimeter of the triangle is betweenA. 14 and 15B. 13 and 14C. 12 and 13D. 11 and 12E. 10 and 11
Answer:
A. Between 14 and 15.
Step-by-step explanation:
Let x be the one leg of the right triangle.
We have been given that the legs of a right triangle are in the ratio of 3 to 1. So, the other leg of the right triangle would be 3x.
We are also told that the length of the hypotenuse of the triangle is √40.
Using Pythagoras theorem, we can set am equation as:
[tex]x^2+(3x)^2=(\sqrt{40})^2[/tex]
Let us solve for x.
[tex]x^2+9x^2=40[/tex]
[tex]10x^2=40[/tex]
[tex]\frac{10x^2}{10}=\frac{40}{10}[/tex]
[tex]x^2=4[/tex]
Take square root of both sides:
[tex]x=\sqrt{4}[/tex]
[tex]x=2[/tex]
The other leg would be [tex]3x\Rightarrow 3\cdot 2=6[/tex].
The perimeter of the triangle would be:
[tex]\text{Perimeter of triangle}=2+6+\sqrt{40}[/tex]
[tex]\text{Perimeter of triangle}=2+6+6.324555[/tex]
[tex]\text{Perimeter of triangle}=14.324555[/tex]
Therefore, the perimeter of the triangle is between 14 and 15 and option A is the correct choice.
Bricklayers use the formula N= 7LH to estimate the number of bricks N needed to build a wall H give. In terms of the length and the number of bricks. b. What is the height of a wall that is 30 feet long and requires 2135 bricks to build?
Answer:
The height of the wall is approximately 10.17 feet.
Step-by-step explanation:
Given,
Length of wall = 30 feet
Total number of bricks = 2135
We have to find out the height of the wall.
To find out the height of wall, we have given the formula;
[tex]N=7LH[/tex]
Where N stands for number of bricks, L stands for length of wall and H stands for height of wall.
So we substitute the given values, we get;
[tex]2135=7\times30\times H\\\\2135=210H\\\\H=\frac{2135}{210}=10.166\approx10.17\ ft\\[/tex]
Hence the height of the wall is approximately 10.17 feet.
The amount of gas that a helicopter uses is directly proportional to the number of hours spent flying. The helicopter flies for 3 hours and uses 33 gallons for fuel. Find the number of gallons of fuel that the helicopter uses to fly for 4 hours.
A) 48 gallonsB) 44 gallonsC) 55 gallonsD) 12 gallons
The helicopter uses 11 gallons of fuel per hour, so for 4 hours, it would use 44 gallons of fuel.
Explanation:The question provides that the number of gallons of fuel used by a helicopter is directly proportional to the number of hours it spends flying. This is a proportionality problem, which can be solved using basic algebraic principles.
Firstly, we know from the question that the helicopter uses 33 gallons of fuel for 3 hours of flight. So, the proportionality constant, or the rate of fuel consumption, is 33 gallons/3 hours = 11 gallons/hour.
To find out how many gallons of fuel the helicopter would consume in 4 hours, we simply multiply the rate of fuel consumption by 4:
11 gallons/hour x 4 hours = 44 gallons.
Therefore, the helicopter would use 44 gallons of fuel to fly for 4 hours.
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Three wholesalers are having special deals on chicken this week. Wholesaler A is selling 10 pounds of chicken for $40.00 , wholesaler B is selling 15 pounds of chicken for $45.00 , and wholesaler C is selling 20 pounds of chicken for $50.00. Which wholesaler has the best price on chicken?
Answer:
Step-by-step explanation:
Wholesaler A is selling 10 pounds of chicken for $40.00. This means that the unit rate at which Wholesaler A is selling the chicken is
40/10 = $4 per pound of chicken.
Wholesaler B is selling 15 pounds of chicken for $45.00. This means that the unit rate at which Wholesaler B is selling the chicken is
45/15 = $3 per pound of chicken.
Wholesaler C is selling 20 pounds of chicken for $50.00. This means that the unit rate at which Wholesaler C is selling the chicken is
50/20 = $2.5 per pound of chicken.
Wholesaler C has the best price because she has the lowest unit rate pound of chicken.
If one side of the triangle is 12 and the opposite angle is 30 degrees, then the diameter of the circumscribed circle is:________
Answer: The required length of the diameter of the circumscribed circle is 24 units.
Step-by-step explanation: Given that one side of a triangle has length 12 units and its opposite angle measures 30 degrees.
We are to find the diameter of the circumscribed circle.
We know that
if a, b, c are the lengths of the three sides of a triangle and A, B, C are the corresponding measures of the opposite angles respectively, then the ratio
[tex]\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}=d,[/tex]
is said to the length of the diameter of the circumscribed circle of the triangle.
According to the given information, we have
a = 12 and A = 30°.
Therefore, the length of the diameter of the circumscribed circle is
[tex]d=\dfrac{a}{\sin A}=\dfrac{12}{\sin 30^\circ}=\dfrac{12}{\frac{1}{2}}=24.[/tex]
Thus, the required length of the diameter of the circumscribed circle is 24 units.
Given the details of a triangle with one of its sides as 12 and opposite angle as 30 degrees, the diameter of the circumscribed circle is twice the length of thegiven side. Since the hypotenuse of the triangle is the diameter of a circumscribed circle, the diameter in this case is 24.
Explanation:The subject of this question is pertaining to geometry, specifically, the relationship between sides and angles in triangles, and circles. In particular, we are dealing with a situation where we have a triangle inscribed in a circle (a triangle with a circumscribed circle), and we are asked to find the diameter of the said circle.
Now, from trigonometry, we know that the sine of any angle in a right triangle is defined as the length of the opposite side divided by the length of the hypotenuse. In this case, you're given that one side of the triangle (which we'll assume is the opposite side) is 12, and its opposite angle is 30 degrees. Since the sine of 30 degrees is 0.5, this means that the hypotenuse of this triangle is actually twice the length of the given side. Thus, the hypotenuse is 2 * 12 = 24.
This is significant because, in any triangle inscribed in a circle, the diameter of the circle is equal to the length of the hypotenuse. Therefore, in this case, the diameter of the circle would be 24.
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39 packs of coffee were sold at a coffee shop, and 45 packs of tea, for which the costumers payed 738. The next day 11 packs of coffee were sold and 13 packs of tea, which made 210 dollars. How much does each package cost
The cost of one pack of coffee is $12 and cost of one tea pack is $6.
Step-by-step explanation:
Let,
Cost of one pack of coffee = x
Cost of one pack of tea = y
According to given statement;
39x+45y=738 Eqn 1
11x+13y=210 Eqn 2
Multiplying Eqn 1 by 11
[tex]11(39x+45y=738)\\429x+495y=8118\ \ \ Eqn\ 3\\[/tex]
Multiplying Eqb 2 by 39
[tex]39(11x+13y=210)\\429x+507y=8190\ \ \ Eqn\ 4[/tex]
Subtracting Eqn 3 from Eqn 4
[tex](429x+507y)-(429x+495y)=8190-8118\\429x+507y-429x-495y=72\\12y=72[/tex]
Dividing both sides by 12
[tex]\frac{12y}{12}=\frac{72}{12}\\y=6[/tex]
Putting y=6 in Eqn 2
[tex]11x+13(6)=210\\11x+78=210\\11x=210-78\\11x=132[/tex]
Dividing both sides by 11
[tex]\frac{11x}{11}=\frac{132}{11}\\x=12[/tex]
The cost of one pack of coffee is $12 and cost of one tea pack is $6.
Keywords: linear equation, subtraction
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Answer:
A pack of coffee=$12
A pack of tea=$6
Step-by-step explanation:
Let the cost of 1 pack of coffee =c
and the cost of 1 pack of tea =t
we can the write the following equations
39c+45t=738...........eqn(1)
11c+13t=738...........eqn(2)
Multiply eqn(1) by 11
429c+495t=8118...........eqn(3)
Multiply eqn(2) by 39
429c+507t=8190...........eqn(4)
eqn(4)-eqn(3)
This implies
12t=72
Dividing through by 12 ,we get
t=6
substituting the value of t into equation two,we obtain
11c+13(6)=210
11c+78=210
11c=210-78
11c=132
Dividing through by 11,we obtain
c=12
What is the 100th term of the sequence with a1 = 222 and d = -5?
-273
-278
717
722
Answer:
-273
Step-by-step explanation:
The n-th term of the sequence is given by ...
an = a1 +d(n -1)
Filling in the given values and doing the arithmetic, we get ...
a100 = 222 +(-5)(100 -1) = 222 -495
a100 = -273
Answer:
-273
Step-by-step explanation:
Machines X and Y work at their respective constant rates. How many more hours does it take machine Y, working alone, to fill a production order of a certain size than it takes machine X, working alone?
1. Machine X and Y, working together, fill a production order of this size in two thirds the time that Machine X does.2. Machine Y, working alone, fills a production order of this size in twice the time that Machine X, working alone, does.
Answer:
1) [tex]\Delta t=-0.5*t_{X}[/tex]
2) [tex]\Delta t=2*t_{X}[/tex]
Step-by-step explanation:
1) X=production rate of X (units/hour)
Y=production rate of Y (units/hour)
C=order (units)
t=working time (hour)
[tex](X+Y)*t_{toghether}=C[/tex]
[tex]X*t_{X}=C[/tex]
[tex]t_{toghether}=2/3*t_{X}[/tex]
Combining:
[tex](X+Y)*2/3*t_{X}=X*t_{X}[/tex]
[tex](X+Y)*2/3=X[/tex]
[tex]Y*2/3=X-2/3*X[/tex]
[tex]Y=2X[/tex]
In time: [tex]t_{Y}=0.5*t_{X}[/tex]
Difference of time: [tex]\Delta t=0.5*t_{X}-t_{X}=-0.5*t_{X}[/tex]
2) [tex]Y*t_{Y}=C[/tex]
[tex]X*t_{X}=C[/tex]
[tex]t_{Y}=2*t_{X}[/tex]
Difference of time: [tex]\Delta t=2*t_{X}-0.5*t_{X}=2*t_{X}[/tex]
At the beginning of the year, the ratio of juniors to seniors in high school X was 3 to 4. During the year, 10 juniors and twice as many seniors transferred to another high school, while no new students joined high school X. If, at the end of the year, the ratio of juniors to seniors was 4 to 5, how many seniors were there in high school X at the beginning of the year?A. 80B. 90C. 100D. 110E. 120
Answer: E. 120
The number of seniors were there in high school X at the beginning of the year = 120
Step-by-step explanation:
Given : At the beginning of the year, the ratio of juniors to seniors in high school X was 3 to 4.
Let the number of juniors be 3x and the number of seniors be 4x.
Since , during the year, 10 juniors and twice as many seniors transferred to another high school, while no new students joined high school X.
i.e. Number of juniors at the end of the year = 3x-10
Number of seniors at the end of the yea = 4x-2(10)=4x-20
At the end of the year, the ratio of juniors to seniors was 4 to 5.
[tex]\Rightarrow\dfrac{3x-10}{4x-20}=\dfrac{4}{5}[/tex]
[tex]\Rightarrow5(3x-10)=4(4x-20)[/tex]
[tex]\Rightarrow15x-50=16x-80[/tex]
[tex]\Rightarrow16x-15x=80-50[/tex]
[tex]\Rightarrow x=30[/tex]
The number of seniors were there in high school X at the beginning of the year = 4(30)=120
Hence, the correct answer is E. 120 .
To find the number of seniors at the high school X at the beginning of the year, calculate based on the given ratio and student transfers.
Explanation:At the beginning of the year:
Let the number of juniors be 3x and the number of seniors be 4x.During the year, 10 juniors and 20 seniors left, making the new ratio of juniors to seniors 4 to 5.Set up an equation using the given information and solve to find that there were 80 seniors at the beginning of the year.Prove that if $w,z$ are complex numbers such that $|w|=|z|=1$ and $wz\ne -1$, then $\frac{w+z}{1+wz}$ is a real number.
Answer:
See proof below
Step-by-step explanation:
Let [tex]r=\frac{w+z}{1+wz}[/tex]. If w=-z, then r=0 and r is real. Suppose that w≠-z, that is, r≠0.
Remember this useful identity: if x is a complex number then [tex]x\bar{x}=|x|^2[/tex] where [tex]\bar{x}[/tex] is the conjugate of x.
Now, using the properties of the conjugate (the conjugate of the sum(product) of two numbers is the sum(product) of the conjugates):
[tex]\frac{r}{\bar{r}}=\frac{w+z}{1+wz} \left(\frac{1+\bar{w}\bar{z}}{\bar{w}+\bar{z}}{\right)[/tex]
=[tex]\frac{(w+z)(1+\bar{w}\bar{z})}{(1+wz)(\bar{w}+\bar{z})}=\frac{w+z+w\bar{w}\bar{z}+z\bar{z}\bar{w}}{\bar{w}+\bar{z}+\bar{w}wz+\bar{z}zw}=\frac{w+z+w+|w|^2\bar{z}+|z|^2\bar{w}}{\bar{w}+\bar{z}+|w|^2z+|z|^2w}=\frac{w+z+\bar{z}+\bar{w}}{\bar{w}+\bar{z}+z+w}=1[/tex]
Thus [tex]\frac{r}{\bar{r}}=1[/tex]. From this, [tex]r=\bar{r}[/tex]. A complex number is real if and only if it is equal to its conjugate, therefore r is real.
a rectangular picture 16 inches by 20 inches has a frame of uniform width. Find the width of the frame if the total area of the picture and the frame is 672 square inches
Answer: 22
Step-by-step explanation:
The area of the picture =l×b = 20×16=320
The length of the frame is 16
If the width is x ( and it's uniform)
672- 320= 352
Area of the frame = 16× x=352
x=352/16
x = 22
Total width is 42 inch.
Given that,
Length of picture is 16 inch and width is 20 inch.Total area with frame is 672 sq in.We need to find the width of frame.According to the scenario, computation of the given data are as follows,
Total area of picture = 16 [tex]\times[/tex] 20 = 320 sq in.
So Frame area = 672 - 320 = 352
Now, length of frame = 16
Let, width of frame = w
So, 16 [tex]\times[/tex] w = 352
W = 352 [tex]\div[/tex] 16
W = 22 inch
So total width is 22 + 20 = 42 inch.
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In a state pick 4 lottery game, a bettor selects four numbers between 0 and 9 and any selected number can be used more than once. Winning the top prize requires that the selected numbers match those and are drawn in the same order.
Do the calculations for this lottery involve the combinations rule or either of the two permutations rules? Why or why not? If not, what rule does apply?
A. The permutation rule (with different items) applies to this problem because repetition is allowed. The permutation rule (with some identical items) and the combination rule cannot be used with repetition.
B. The combination and permutations rules do not apply because repetition is allowed and numbers are selected with replacement. The factorial rule applies to this problem.
C. The combination and permutations rules do not apply because repetition is allowed and numbers are selected with replacement . The multiplication counting rule applies to this problem.
D. The combination rule applies to this problem because the numbers are selected with replacement. Neither of the permutations rules allows replacement.
E. The permutation rule (with some identical items) applies to this problem because repetition is allowed The permutation rule (with different items) and the combination rule cannot be used with repetition.
Answer:
the answer is na
Step-by-step explanation:
Final answer:
The permutation rule (with different items) applies to this problem because repetition is allowed. The permutation rule (with some identical items) and the combination rule cannot be used with repetition.
Explanation:
The correct answer is A. The permutation rule (with different items) applies to this problem because repetition is allowed. The permutation rule (with some identical items) and the combination rule cannot be used with repetition. The calculations for this lottery game involve the permutation rule because the bettor selects four numbers between 0 and 9, and any selected number can be used more than once. The permutation rule allows for repetition and ensures that the order of the selected numbers is taken into account when determining the probability of winning the top prize.