the LCM of 21 and 105 is 105.
Answer:
The answer to your question is 105
Step-by-step explanation:
To get the least common factor dividing both numbers by the prime numbers starting from 2.
But these numbers are not multiples of 2, try 3
21 105 3
These numbers are not multiples of 3 let's try 5
7 35 5
These numbers are multiples of 7 7 7 7
1 1
The LCM is the product of the prime numbers = 3 x 5 x 7 = 105
Three lattice points (points with integer coordinates) are chosen at random with replacement (meaning you can select the same point more than once) in the interior of the square defined by $-99 \le x \le 100$ and also $-99 \le y \le 100$. The probability that the area of the triangle thus formed (which may be degenerate) is an integer can be expressed in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
This is a combinatorial geometry problem involving the probability of forming a triangle with an integer area from randomly selected lattice points within a square grid. It requires analyzing the properties of lattice point triangles and applying combinatorial mathematics.
Explanation:The problem given discusses the selection of random points within a square on a coordinate system to form a triangle, and it questions the probability of the triangle's area being an integer. This involves combinatorics, geometry, and number theory within the field of mathematics. To approach the problem of calculating the probability that the area is an integer, one must consider the distribution of lattice points and use combinatorial analysis.
Area of a triangle with vertices at lattice points can be expressed using the formula derived from the determinant of a matrix. The triangle's area, if non-degenerate, can be half of the odd integer, which upon doubling to remove the half, leaves an integer area. Thus, the problem reduces to a combinatorial question concerning lattice points, selected at random, and forming a triangle whose twice the area is an odd integer.
However, a direct count of the favorable vs. total possibilities may be complex due to the large number of possible point combinations and the chance of degenerate triangles (where the area is zero). Thus, a systematic probabilistic and combinatorial approach is necessary to resolve this question, likely involving a careful analysis of the geometric configurations and properties of integer triangles on a lattice grid.
PLEASE HELP
A class of 20 students is visiting the fair today. Each student is 16 years old or younger and will participate in either the Pie Eating Contest or the Chili Cookoff. If all 20 students joined the same event, how would the shape of the histogram change compared to the original?
Histogram with title Pie Eating Contest, horizontal axis labeled Age Group (year) with bins 0 to 19, 20 to 39, 40 to 59, and 60 to 79 and vertical axis labeled Number of People with values from 0 to 60 at intervals of 10. The first bin goes to 20, the second goes to 40, the third goes to 60, and the last goes to 50. Histogram with title Chili Cookoff, horizontal axis labeled Age Group (year) with bins 0 to 19, 20 to 39, 40 to 59, and 60 to 79 and vertical axis labeled Number of People with values from 0 to 60 at intervals of 10. The first bin goes to 30, the second goes to 50, the third goes to 40, and the last goes to 10.
The Chili Cookoff would be more skewed.
The Pie Eating Contest would be more skewed.
The Chili Cookoff would be more symmetrical.
The Pie Eating Contest would be less symmetrical.
Answer:
A
Step-by-step explanation:
The interval of 0-19 will increase by 20 of either Hay Bale Toss or Pie Eating Contest.
What is histogram?A histogram is a visual representation of statistical data that makes use of rectangles to illustrate the frequency of data items in a series of equal-sized numerical intervals. The independent variable is represented along the horizontal axis and the dependent variable is plotted along the vertical axis in the most popular type of histogram.
Given that, a class of 20 students is visiting the fair today. Each student is 16 years old or younger and will participate in either the Pie Eating Contest or the Chili Cookoff.
Number of students visiting the fair = 20
Age group = 0-16
In the histogram, this age group falls in the first interval which is 0-19.
If all 20 students joined the Hay Bale Toss then the bar of 0-19 of Hay Bale Toss will increase from 40 to 60 and the graph of Pie Eating Contest will remain the same.
If all 20 students joined the Pie Eating Contest then the bar of 0-19 of Hay Bale Toss will increase from 20 to 40 and the graph of Hay Bale Toss will remain the same.
Therefore, only one if the graph's interval of 0-19 will increase by 20 whereas the other graph would remain the same.
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A new pencil is 19 centimeters long. Levi uses it for a month. The pencil now measures 8 centimeters.
The pencil is centimeters shorter now than when it was new.
Final answer:
The pencil is 11 centimeters shorter now than when it was new, which is determined by subtracting the current length (8 centimeters) from the original length (19 centimeters).
Explanation:
The subject of this question is Mathematics. The question involves calculating the length by which a pencil has become shorter, which is a basic arithmetic problem dealing with subtraction.
To find how much shorter the pencil is now compared to when it was new, we subtract the current length of the pencil from its original length. Given that the new pencil was 19 centimeters long and now measures 8 centimeters, the calculation would be:
Original length - Current length = Length difference
19 cm - 8 cm = 11 cm
Therefore, the pencil is 11 centimeters shorter now than when it was new.
The unit of measurement is particularly important here to provide clarity and accuracy to the measurement. For example, while measuring the length of a classroom, you might use meters or feet, but for small objects like a pencil, centimeters or inches are more appropriate.
43 less then the product of 159 and n
Answer:
159n - 43
Step-by-step explanation:
Write as an equation
159 * n - 43
Multiply
159n - 43
Hope this helps :)
Every force has both of what
Answer:
A force is a vector quantity. As learned in an earlier unit, a vector quantity is a quantity that has both magnitude and direction. To fully describe the force acting upon an object, you must describe both the magnitude (size or numerical value) and the direction.
Step-by-step explanation:
factor the Expression 42x+49
Answer:
[tex]42x + 49 \\ = 7(6x + 7)[/tex]
hope this helps..!
An adventure game has a number cube with 12 equal-size faces. Each face is labeled with a number from 1 to 12.
What is the probability of rolling a 4 or a multiple of 5?
Enter your answer as a fraction, in simplified form, in the box.
Need the exact answer.
The probability of getting the number 4 and the multiple of 5 is 1 / 4.
What is probability?Probability is defined as the ratio of the number of favourable outcomes to the total number of outcomes in other words the probability is the number that shows the happening of the event.
Probability = Number of favourable outcomes / Number of samples
Given that an adventure game has a number cube with 12 equal-size faces. Each face is labelled with a number from 1 to 12.
The probability will be calculated as:-
Number of favourable outcomes = { 4, 5 , 10 } = 3
Number of samples = 12
Probability = 3 / 12
Probability = 1 / 4
Therefore, the probability of getting the number 4 and the multiple of 5 is 1 / 4.
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Margot used
1
2
pounds of turkey to make
2
3
liters of turkey chili. If Margot's recipe is for 1 liter of turkey chili, how many pounds of turkey will she need to make 4 times the recipe?
Margot needs 3/4 pounds of turkey to make 1 liter of turkey chili. To make 4 liters of turkey chili, which is 4 times the recipe, she will need 3 pounds of turkey.
Margot used 1/2 pound of turkey to make 2/3 liters of turkey chili. To find out how much turkey she will need for 4 liters of turkey chili (4 times the 1-liter recipe), we need to calculate the amount of turkey needed per liter first and then multiply it by 4.
First, we determine the amount of turkey needed for 1 liter by setting up a proportion. Since 1/2 pound of turkey makes 2/3 liters, we have:
1/2 pound / (2/3) liter = x pounds / 1 liter
By cross-multiplication, we find that x = (1/2) * (3/2) = 3/4. Therefore, Margot needs 3/4 pounds of turkey for 1 liter of chili.
To make 4 times the recipe, or 4 liters, Margot will need 4 * 3/4 pounds of turkey = 3 pounds of turkey.
which of the following choices are the angle and side lengths of the given triangle
Answer:
The correct answer is D. 30°, 60°, 90°; 1, √3, 2
Step-by-step explanation:
Let's find out the solution to this question:
Arc cos (β) = √3/2 = 1.732/2 = 0.866
β = 30°
Let's recall that the ratio of the sides of a triangle 90 - 60 - 30 is:
1 : 2 : √3
β = 30°, α = 60°, r = 2, h = 1, k = √3
The correct answer is D. 30°, 60°, 90°; 1, √3, 2
(8x-6)(7x+6) multiple together and simplify
Answer:
56x² + 6x - 36
Step-by-step explanation:
(8x-6)(7x+6) = 8x*(7x + 6) - 6*(7x + 6)
= 8x*7x + 8x*6 - 6*7x - 6*6
=56x² + 48x - 42x - 36
= 56x² + 6x - 36
One of the roots of the equation 5x2−36x+t=0 is five times as big as the other root. Find the value of t.
Answer:
t = 36
Step-by-step explanation:
If one of the roots is "a", the equation can be factored as ...
(5x -a)(x -a) = 0
5x^2 -6ax +a^2 = 0
Comparing terms to the given equation, we see that ...
-6ax = -36x
a = 6 . . . . . . . . divide by -6
Then ...
a^2 = t
36 = t . . . . . . . substitute 6 for a
_____
The roots are 6 and 6/5.
Emma collected 72 apples from the orchard and put an equal number of apples into 6 baskets. Mrs. Doolan wants 3 baskets of apples. How many apples will Mrs. Doolan receive?
Answer:
36 apples
Step-by-step explanation:
First, divide 72 by 6 to find out how many apples are in 1 basket.(Answer 12)
There will be 12 apples in one basket.
Mrs. Doolan wants 3 baskets so multiply 12 by 3 to get 36.
72/6= 12
12X3=36
Answer:36 apples
Step-by-step explanation: dividing that will each be 12 apples in a basket. mrs. doolan will recieve three baskets so 12x3. that will equal 36
What slope does a horizontal line have?
Answer:
Step-by-step explanation:
Horizontal
Answer:
slope = 0
Step-by-step explanation:
A horizontal line is parallel to the x- axis.
Since the x- axis has a slope of zero then the slope of a horizontal line is 0
After transforming f(x) = 2x² +4x + 3 into vertex form, the vertex is easily identifiable. Which ordered pair is the vertex?
(-1, 1)
(0, 3)
(1, 1)
(-3, 0)
The vertex is [tex](-1,1)[/tex]
Explanation:
The equation is [tex]f(x)=2x^{2} +4x+3[/tex]
To find the vertex, we need the equation in the form of [tex]f(x)=a (x-h)^{2}+k[/tex]
Dividing each term by 2 in the equation [tex]f(x)=2x^{2} +4x+3[/tex]
[tex]f(x)=2(x^{2} +2x+\frac{3}{2} )[/tex]
Now, completing the square by adding and subtracting 1, we get,
[tex]f(x)=2(x^{2} +2x+1-1+\frac{3}{2} )[/tex]
The first three terms can be written as [tex](x+1)^{2}[/tex],
[tex]f(x)=2[(x+1)^{2}+\frac{1}{2} ][/tex]
Multiplying 2 into the bracket, we get,
[tex]f(x)=2(x+1)^{2} +1[/tex]
This equation is of the form [tex]f(x)=a (x-h)^{2}+k[/tex]
Now, we shall find the vertex [tex](h,k)[/tex]
Thus, [tex]h=-1[/tex] and [tex]k=1[/tex]
Thus, the vertex is [tex](-1,1)[/tex]
Graph the line that represents a proportional relationship between y and x where the unit rate of change of y with respect to x xx is 0.4. point, 4, point In other words, a change of 1 unit in x corresponds to a change of 0.4, point, 4 units in y.
Answer:
The graph is attached.
Step-by-step explanation:
The graph of the line with the slope [tex]m[/tex] and the y-intercept [tex]b[/tex], has the form
[tex]y=mx+b[/tex]
In our case the rate of change [tex]m[/tex] is [tex]0.4[/tex]; therefore we have the equation
[tex]y=0.4x[/tex]
The graph of which is attached.
P.S: we have set [tex]b=0[/tex] by default, because we are not given any information for it.
The line that represents a proportional relationship between y and x where the unit rate of change (or slope) is 0.4 can be plotted on a graph. Start from the origin and for each increase of 1 in x, y increases by 0.4. Plot these points and draw a straight line which represents this proportional relationship.
Explanation:To graph the line that represents a proportional relationship between y and x where the unit rate of change of y with respect to x is 0.4, you can start by creating a plot with an x- and y-axis. The y-axis represents the dependent variable and the x-axis represents the independent variable.
In this case, we're looking at a linear relationship, which means the line produced on the graph will be straight. This kind of straight-line graph typically uses the equation y = mx, where m is the slope (or unit rate of change) and x is the independent variable.
For this graph, the slope or unit rate of change is 0.4. This means that with every increase of 1 in x, y increases by 0.4. Plot some points on the graph starting from the origin (0,0). If x is 1, y will be 0.4 (1*0.4). If x is 2, y will be 0.8 (2*0.4). Keep plotting these points and connect them to draw the line. This line demonstrates the proportional relationship between x and y.
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please help me guys, I'm so confused
[tex]\frac{8}{35}[/tex] of all claims were paid in the second month.
Step-by-step explanation:
Step 1; In the first month [tex]\frac{3}{7}[/tex] of all expenses were paid. So to find the expenses that were not paid in the first month we subtract [tex]\frac{3}{7}[/tex] from 1 i.e. 1-[tex]\frac{3}{7}[/tex]= [tex]\frac{4}{7}[/tex]. So in the first month [tex]\frac{4}{7}[/tex] of total expenses were not paid.
Step 2; In the second month [tex]\frac{2}{5}[/tex] of the remaining claims were paid i.e. [tex]\frac{2}{5}[/tex] of the remaining claims which is [tex]\frac{4}{7}[/tex] of the total claims. So to find out the number of claims paid in the second month it's just a case of multiplying [tex]\frac{2}{5}[/tex] with [tex]\frac{4}{7}[/tex].
Fraction of all claims paid in second month = [tex]\frac{2}{5}[/tex] * [tex]\frac{4}{7}[/tex]= [tex]\frac{8}{35}[/tex].
Example; If $ 70,000 was the total claim amount, [tex]\frac{3}{7}[/tex] of that which equals $30,000 was paid in the first month. So $40,000 was still to be paid. In the second month [tex]\frac{2}{5}[/tex] of $40,000 was paid which equals $16,000 paid in the second month which is [tex]\frac{8}{35}[/tex] of total claims i.e. $70,000
An artist uses 49.4 centimeters of decorative molding to make a square picture frame . How long is each side of the frame
Answer:
Each side of the frame is 12.35 centimeters long.
When two six-sided dice are rolled, there are 36 possible outcomes. Find the probability that the sum is not 5.
Answer:
[tex]p(Not\ 5) = \frac{8}{9}[/tex]
Step-by-step explanation:
Given:
Txo six-sided dice are rolled.
Total number of outcomes n(S) = 36
We need to find the probability that the sum is not equal to 5 p(Not 5).
Solution:
Using probability formula.
[tex]P(E)=\frac{n(E)}{n(S)}[/tex] ----------------(1)
Where:
n(E) is the number of outcomes favourable to E.
n(S) is the total number of equally likely outcomes.
The sum of two six-sided dice roll outcome is equal to 5 as.
Outcome as 5: {(1,4), (2,3), (3,2), (4,1)}
So, the total favourable events n(E) = 4
Now, we substitute n(E) and n(s) in equation 1.
[tex]P(5)=\frac{4}{36}[/tex]
[tex]p(5) = \frac{1}{9}[/tex]
Using formula.
[tex]p(Not\ E) + p(E) = 1[/tex]
[tex]p(Not\ 5) + p(5) = 1[/tex]
Now we substitute p(5) in above equation.
[tex]p(Not\ 5) + \frac{1}{9} = 1[/tex]
[tex]p(Not\ 5) = 1-\frac{1}{9}[/tex]
[tex]p(Not\ 5) = \frac{9-1}{9}[/tex]
[tex]p(Not\ 5) = \frac{8}{9}[/tex]
Therefore, the sum of two six-sided dice roll outcome is not equal to 5.
[tex]p(Not\ 5) = \frac{8}{9}[/tex]
Carol's monthly take home pay is 1500$. She spends $250 a month on food. What is the ratio of food costs to take home dollars?
Answer:
The ratio of food costs to take home is 1 : 6.
Step-by-step explanation:
Given:
Carol's monthly take home pay is 1500$. She spends $250 a month on food.
Now, to find the ratio of food costs to take home pay.
Take home pay = $1500.
Cost of food = $250.
So, getting the ratio:
Cost of food : take home pay
$250 : $1500
[tex]=\frac{250}{1500}[/tex]
[tex]=\frac{1}{6}[/tex]
[tex]=1:6[/tex]
Therefore, the ratio of food costs to take home is 1 : 6.
Final answer:
Carol's food costs to take home pay ratio is 1:6, meaning for every six dollars she takes home, she spends one dollar on food.
Explanation:
To calculate the ratio of Carol's food costs to her take home pay, we compare the amount she spends on food to her total monthly take home pay. This can be represented by the ratio 250:1500 or simplified by dividing both numbers by the common factor of 250, which results in the simplified ratio of 1:6. Hence, for every dollar Carol takes home, she spends about one-sixth of it on food.
The area of a triangle is 15x^4+3x^3+4x^2-x-3 square meters. The length of the base of the triangle is 6x^2-2 meters. What is the height of the triangle?
Answer:
The height of the triangle is: [tex]H = (5x^2 + x +3)[/tex]
Step-by-step explanation:
Here, the area the triangle is given as: [tex]15x^4 + 3x^3 + 4x^2 - x - 3[/tex]
Also, base of the triangle = [tex](6x^2 -2)[/tex]
Let us assume the height of the triangle = H units
Now, AREA of TRIANGLE = [tex]\frac{1}{2} \times B \times H[/tex]
[tex]\implies (15x^4+3x^3+4x^2-x-3 ) = \frac{1}{2} \times ( 6x^2-2) \times H\\\implies H =\frac{ (15x^4+3x^3+4x^2-x-3 )}{(3x^2-1)}[/tex]
Solving by LONG DIVISION, we get:
[tex]H = (5x^2 + x +3)[/tex]
Hence, the height of the triangle is: [tex]H = (5x^2 + x +3)[/tex]
If f(x)=2/3x+14, then find the value of f(12)
[tex]\dotfill[/tex]
To find the value of the given function, substitute x = 12 instead of x and evaluate.
Given function:
[tex]\sf f(x)=\cfrac{2}{3} x+14[/tex]
Substitute 12 for x:
[tex]\sf f(12)=\cfrac{2}{3}\times12+14[/tex]
Convert 12 to a fraction to make multiplying easier:
[tex]\sf f(12)=\cfrac{2}{3}\times\cfrac{12}{1}+14[/tex]
[tex]\sf f(12)=\cfrac{2\times12}{3\times1}+14[/tex]
[tex]\sf f(12)=\cfrac{24}{3}[/tex]
[tex]\sf f(12)=8[/tex]
>>> Therefore, the value of the function is 8.
The value of [tex]\( f(x) = \frac{2}{3}x + 14 \)[/tex] is 22.
To find the value of f(12), you simply need to substitute (x = 12) into the function f(x) and evaluate it:
Given: [tex]\( f(x) = \frac{2}{3}x + 14 \)[/tex]
Substituting x = 12:
[tex]\[ f(12) = \frac{2}{3} \times 12 + 14 \][/tex]
Now, let's calculate:
[tex]\[ f(12) = \frac{2}{3} \times 12 + 14 \]\[ = \frac{2}{3} \times 12 + 14 \]\[ = 8 + 14 \]\[ = 22 \][/tex]
So, [tex]\( f(12) = 22 \)[/tex].
What is mZLQK ?
Enter your answer in the box.
Dº
Answer:
45
Step-by-step explanation:
The sum of angle in a straight line is 180
=> 3n + (4n - 15) + 90 = 180
3n + 4n + 75 = 180
7n = 180 - 75 = 105
n = 15
Angle ZLQK = 4n - 15 = 4*15 - 15 = 60 - 15 = 45
What is the answer to both of these questions? Please help
Answer:
28) 120 x 20y
30) 3w x 36
Step-by-step explanation:
28) 20 x (6 x y)
20 x 6 = 120
y x 20 = 20 y
120 x 20y
30) (w x 12) x 3
3 x w = 3w
12 x 3 = 36
3w x 36
Answer:
120 x 20y 3w x 36
Step-by-step explanation:
Distributive Property
Which term best describes the following statement?
You are looking at a map of the United States, and it is
assumed that you already know which direction is north.
OA. Theorem
OB. Conjecture
O C. Common notion
O D. Proof
Theorem .
Basically you already know this, and it can also be proven.
Emma ran 3 1/2 km while Grace ran 380 m who ran further
Answer:
Emma
Step-by-step explanation:
Emma ran (3 1/2)×(1000 m) = 3500 m compared to Grace's 380 m. 3500 m is farther, so Emma ran farther.
The length of a rectangle is 8 times it's width. The perimeter is 126 cm. What is the length of the rectangle?
Answer:
width: 7 cm
length: 56 cm
Step-by-step explanation:
Let:
x=width
8x=length
P=2l+2w
126=2x + 2(8x)
126=2x+16x
18x=126
x=7
width: 7 cm
length: 56 cm
hope this helps!!
I WILL GIVE BRAINLIEST
Answer:
- 6
Step-by-step explanation:
its negative so I divided —12 by 2 to get -6 also You have to get the F by itself so you have to divide the sides by 2 but bringing the negative is important
F(x) =2x^2+4x+5
Find: f(a)
Answer:
F(a) =2a^2+4a+5
Step-by-step explanation:
just put x= a
a ladder touches a wall 15ft off the ground, and the base of the ladder is 8 feet from the base of the wall. how long is the ladder
The length of ladder is 17 feet
Solution:
Given that, ladder touches a wall 15 ft off the ground, and the base of the ladder is 8 feet from the base of the wall
To find: length of ladder
The ladder, wall and ground forms a right angled triangle
Where, ladder is "hypotenuse" and "ground' forms the base
The figure is attached below
ABC is a right angled triangle
AC = length of ladder = ?
AB = height of wall = 15 feet
BC = distance between base of ladder and base of wall = 8 feet
Apply pythogoras theorem for right angled triangle
Pythagorean theorem, states that the square of the length of the hypotenuse is equal to the sum of squares of the lengths of other two sides of the right-angled triangle.
By above theorem,
[tex]AC^2 = AB^2+BC^2[/tex]
Substituting the values we get,
[tex]AC^2 = 15^2 + 8^2\\\\AC^2 = 225 + 64\\\\AC^2 = 289\\\\\text{Take square root on both sides }\\\\AC = \sqrt{289}\\\\AC = 17[/tex]
Thus length of ladder is 17 feet
1. a 3-ounce can of tomato sauce costs $1.68. in cents, what is the price per ounce ?
2. how many cds can edward buy for $14 each and spend the same amount he would to buy 60 cds for $7 each ?
Answer: 1) 56 cents. 2) 30 CDs
Step-by-step explanation:
1.68 ÷ 3 = 56 cents
60 x 7 = $420
$420/ 14 = 30 CDs
The price per ounce of tomato sauce is 56 cents. Edward can buy 30 CDs for $14 each with the same amount that would buy 60 CDs for $7 each.
To calculate the price per ounce, we take the total cost of the can in cents, which is $1.68 or 168 cents, and divide it by the number of ounces, which is 3 ounces. Thus, the price per ounce is: 168 cents / 3 ounces = 56 cents per ounce.
Edward wants to spend the same amount to buy CDs at $14 each as he would buying 60 CDs at $7 each. The total cost for 60 CDs at $7 each is: 60 CDs x $7/CD = $420. We divide this total cost by the price of one CD at $14: $420 / $14/CD = 30 CDs. Therefore, Edward can buy 30 CDs for $14 each with the same amount of money.