The dimensions of a rectangular prism are shown. A new prism is created by changing only the height of the prism by a scale factor of two. Select the statement that is true. Responses The surface area of the new prism is twice the surface area of the original prism. The surface area of the new prism is twice the surface area of the original prism. The surface area of the new prism increased by more than a scale factor of two. The surface area of the new prism increased by more than a scale factor of two. The surface area of the new prism increased by less than a scale factor of two. The surface area of the new prism increased by less than a scale factor of two. The surface area of the new prism decreases.

These two numbers are -4, 2. The smallest possible product is -8.

To find a pair of numbers that satisfy the given conditions, we can use algebra. Let x be the first number and y be the second number. According to the problem, one number is 8 less than twice a second number. This can be written as:
x = 2y - 8
We need to find the product of these two numbers, which is x*y. Substituting the value of x from the equation above, we get:
x*y = (2y - 8)*y
= 2y^2 - 8y
To find the smallest possible product, we need to minimize this expression. We can do this by finding the vertex of the parabola represented by this equation. The vertex of a parabola in the form ax^2 + bx + c is given by (-b/2a, f(-b/2a)). In this case, a = 2, b = -8, and c = 0. So, the vertex is:
(-b/2a, f(-b/2a)) = (-(-8)/(2*2), f(-(-8)/(2*2)))
= (2, f(2))
Substituting y = 2 into the equation for the product, we get:
x*y = 2(2)^2 - 8(2)
= 8 - 16
= -8
So, the smallest possible product is -8. To find the pair of numbers that give this product, we can substitute y = 2 into the equation for x:
x = 2y - 8
= 2(2) - 8
= -4
Therefore, the pair of numbers are -4 and 2.

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300  kilowatt-hours per month does one have to return to the grid to break even.

To find out how many kilowatt-hours per month one has to return to the grid to break even, we need to set up an equation and solve for x, where x is the number of kilowatt-hours returned to the grid.

The equation would be:

24 = 0.08x

To solve for x, we need to isolate the variable on one side of the equation. We can do this by dividing both sides of the equation by 0.08:

24/0.08 = x

x = 300

Therefore, one would have to return 300 kilowatt-hours per month to the grid to break even.

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Check the picture below.

[tex]\begin{array}{llll} f(x)~from\\\\ x_1 ~~ to ~~ x_2 \end{array}~\hfill slope = m \implies \cfrac{ \stackrel{rise}{f(x_2) - f(x_1)}}{ \underset{run}{x_2 - x_1}}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array} \\\\[-0.35em] ~\dotfill\\\\ \begin{cases} x_1=0\\ x_2=2 \end{cases}\implies \cfrac{f(2)-f(0)}{2 - 0}\implies \cfrac{18-12}{2}\implies \cfrac{6}{2}\implies \text{\LARGE 3}[/tex]

The partial derivatives of f with respect to u and v at (u,v) = (-2, -2) are ∂f/∂u(-2,-2) = 224 and ∂f/∂v(-2,-2) = -1056.

To evaluate the partial derivatives ∂f/∂u and ∂f/∂v at (u,v) = (-2, -2) using the Chain Rule, we need to find the derivatives of f with respect to x, y, and z, and then multiply them by the derivatives of x, y, and z with respect to u and v.

First, let's find the derivatives of f with respect to x, y, and z:
∂f/∂x = 3x^2
∂f/∂y = z^2
∂f/∂z = 2yz

Next, let's find the derivatives of x, y, and z with respect to u and v:
∂x/∂u = 2u
∂x/∂v = 1
∂y/∂u = 1
∂y/∂v = 2v
∂z/∂u = 4v
∂z/∂v = 4u

Now, we can use the Chain Rule to find the partial derivatives of f with respect to u and v:
∂f/∂u = (3x^2)(2u) + (z^2)(1) + (2yz)(4v)
∂f/∂v = (3x^2)(1) + (z^2)(2v) + (2yz)(4u)

Finally, we can plug in the values of u and v to find the partial derivatives at (u,v) = (-2, -2):
∂f/∂u(-2,-2) = (3((-2)^2 + (-2))^2)(2(-2)) + ((4(-2)(-2))^2)(1) + (2((-2)+(-2)^2)(4(-2)))
= (3(0)^2)(-4) + (16^2)(1) + (2(2)(-8))
= 0 + 256 - 32
= 224

∂f/∂v(-2,-2) = (3((-2)^2 + (-2))^2)(1) + ((4(-2)(-2))^2)(2(-2)) + (2((-2)+(-2)^2)(4(-2)))
= (3(0)^2)(1) + (16^2)(-4) + (2(2)(-8))
= 0 - 1024 - 32
= -1056

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The approximate value of x to the nearest tenth of a degree is 36.9 degrees.

To find x to the nearest tenth of a degree, we can use trigonometric ratios for right triangles. First, we need to determine which trigonometric ratio to use based on the given information.

If we are given the opposite side and the adjacent side of the right triangle, we can use the tangent ratio:

tan(x) = opposite/adjacent

If we are given the opposite side and the hypotenuse, we can use the sine ratio:

sin(x) = opposite/hypotenuse

If we are given the adjacent side and the hypotenuse, we can use the cosine ratio:

cos(x) = adjacent/hypotenuse

Once we have determined the appropriate trigonometric ratio, we can plug in the given values and solve for x. To find x to the nearest tenth of a degree, we can use a calculator to find the approximate value of x and then round to the nearest tenth.

For example, if we are given a right triangle with an opposite side of 3 and an adjacent side of 4, we can use the tangent ratio to find x:

tan(x) = 3/4

x = tan^-1(3/4)

Using a calculator, we find that x is approximately 36.87 degrees. To the nearest tenth of a degree, x is approximately 36.9 degrees.

Therefore, the approximate value of x to the nearest tenth of a degree is 36.9 degrees.

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Answer:

Step-by-step explanation:

1+2x)5

First you minus the 1 with the 5

Which you'll get a four then divide it by 2

Which you'll get x=2

But then times it by 5

Then you get y=10

Answer:

[tex]\dfrac{d}{dx}\left(\sqrt{1\:+\:2x}\right)5 = \boxed{\dfrac{5}{\sqrt{1+2x}}}[/tex]

Step-by-step explanation:

Given [tex]y = \left(\sqrt{1\:+\:2x}\right)5[/tex]

we are asked to find [tex]\dfrac{dy}{dx}[/tex]

[tex]\dfrac{dy}{dx} = \dfrac{d}{dx}\left(\sqrt{1\:+\:2x}\right)5\\\\= 5\dfrac{d}{dx}\left(\sqrt{1+2x}\right)\\\\[/tex]

Find [tex]\dfrac{d}{dx}\left(\sqrt{1+2x}\right)[/tex]:
[tex]Let \;u = 1 + 2x\\\\f(u) = \sqrt(u)\\\\[/tex]

[tex]\mathrm{Apply\:the\:chain\:rule}:\quad \dfrac{df\left(u\right)}{dx}=\dfrac{df}{du}\cdot \dfrac{du}{dx}[/tex]

[tex]= \dfrac{d}{du}\left(\sqrt{u}\right)\dfrac{d}{dx}\left(1+2x\right)[/tex]

[tex]\dfrac{d}{du}\left(\sqrt{u}\right) = \dfrac{d}{du}\left(u^{\dfrac{1}{2}}\right)\\\\= \dfrac{1}{2}u^{\dfrac{1}{2}-1}\\\\= \dfrac{1}{2\sqrt{u}}\\\\\\[/tex]

Substitute back u = 1 + 2x
[tex]= \dfrac{1}{2\sqrt{1+2x}}[/tex]

[tex]\dfrac{d}{dx}(1 + 2x) =\dfrac{d}{dx}(1)} + \dfrac{d}{dx}{2x}\\\\= 0 + 2 \\\\= 2\\[/tex]

Therefore
[tex]\dfrac{dy}{dx} = \dfrac{d}{dx}\left(\sqrt{1\:+\:2x}\right)5\\\\= 5\dfrac{d}{dx}\left(\sqrt{1+2x}\right)\\\\[/tex]

[tex]= 5\cdot \dfrac{1}{2\sqrt{1+2x}}\cdot \:2\\\\= 5\cdot \dfrac{1}{\sqrt{1 + 2x}}\\\\=\dfrac{5}{\sqrt{1+2x}}[/tex]

The total surface area of the cube is 3.84 m². The solution has been obtained by using the area of cube.

The total surface area of a given cube is said to be equal to the sum of all the surface areas of the cube's faces, according to the definition of surface area. Given that the cube has six faces, its total surface area will be equal to the sum of its six faces.

We are given a cube with side length 0.8 metres.

We know that total surface area of a cube is given by 6a².

Now, by substituting a = 0.8 in the formula, we get

⇒Total surface area of cube = 6 (0.8)²

⇒Total surface area of cube = 6 (0.64)

⇒Total surface area of cube = 3.84 m²

Hence, the total surface area of the cube is 3.84 m².

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Answer:

(5,7)



x = 5

y = 7

The expression is (+(13)/(18))+(-(1)/(3)). To simplify this expression, we need to combine like terms. First, we can combine the fractions with different denominators. The LCD (Least Common Denominator) for this expression is 18. Therefore, we need to convert each fraction to an equivalent fraction with a denominator of 18.

For the first fraction, (+(13)/(18)), 13/18 can be reduced to 1/2. To get this, multiply both the numerator and denominator by 2, resulting in 13/18 = 2/4 = 1/2.

For the second fraction, +(-(1)/(3)), 1/3 can be reduced to 6/18. To get this, multiply both the numerator and denominator by 6, resulting in 1/3 = 6/18.

Now that all the fractions have a common denominator, we can add the two fractions together. 1/2 + 6/18 = 8/18. Therefore, the simplified form of the expression is 8/18.

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21. tan^2 t/sec t = sec t-cos t
First, we can simplify the left-hand side of the equation by dividing tan^2 t by sec t. Remember that tan t = sin t/cos t and sec t = 1/cos t:
tan^2 t/sec t = (sin^2 t/cos^2 t)/(1/cos t) = sin^2 t/cos t
Now we can subtract cos t from both sides of the equation:
sin^2 t/cos t - cos t = sec t - cos t
Next, we can multiply both sides of the equation by cos t to get rid of the fraction:
sin^2 t - cos^2 t = sec t*cos t - cos^2 t
Finally, we can use the Pythagorean identity sin^2 t + cos^2 t = 1 to simplify the left-hand side of the equation:
1 - cos^2 t - cos^2 t = sec t*cos t - cos^2 t
0 = sec t*cos t - 2*cos^2 t
22. cot^2 t/csc t = csc t-sin t
First, we can simplify the left-hand side of the equation by dividing cot^2 t by csc t. Remember that cot t = cos t/sin t and csc t = 1/sin t:
cot^2 t/csc t = (cos^2 t/sin^2 t)/(1/sin t) = cos^2 t/sin t
Now we can subtract sin t from both sides of the equation:
cos^2 t/sin t - sin t = csc t - sin t
Next, we can multiply both sides of the equation by sin t to get rid of the fraction:
cos^2 t - sin^2 t = csc t*sin t - sin^2 t
Finally, we can use the Pythagorean identity sin^2 t + cos^2 t = 1 to simplify the left-hand side of the equation:
1 - sin^2 t - sin^2 t = csc t*sin t - sin^2 t
0 = csc t*sin t - 2*sin^2 t
24. (1−sin θ)/cos θ = sec θ−tan θ
First, we can simplify the left-hand side of the equation by dividing 1-sin θ by cos θ:
(1−sin θ)/cos θ = (1/cos θ) - (sin θ/cos θ) = sec θ - tan θ
So the equation is already simplified and both sides are equal.
25. (sin t/csc t) + (cos t/sec t) = 1
First, we can simplify the left-hand side of the equation by dividing sin t by csc t and cos t by sec t. Remember that csc t = 1/sin t and sec t = 1/cos t:
(sin t/csc t) + (cos t/sec t) = (sin t/(1/sin t)) + (cos t/(1/cos t)) = sin^2 t + cos^2 t
Now we can use the Pythagorean identity sin^2 t + cos^2 t = 1 to simplify the left-hand side of the equation:
1 = 1
So the equation is already simplified and both sides are equal.

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From the given equation the value of r is 8.

A mathematical statement known as an equation is made up of two expressions joined together by the equal sign. A formula would be 3x - 5 = 16, for instance. When this equation is solved, we discover that the value of the variable x is 7.

To solve this equation for r, we need to isolate r on one side of the equation. First, we add 14 to both sides of the equation:

4r-14+14 = (9/4)r+14

4r = (9/4)r+14

Then, we subtract (9/4)r from both sides:

4r - (9/4)r = (9/4)r+14 - (9/4)r

(5/4)r = 14

Finally, divide both sides of the equation by (5/4) to isolate r:

r = 14/(5/4) = 8

Therefore, the value of r is 8.

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We have:

Find the Range.

Substitute to F(x):

x = -1, x = 0, x = 1 and x = 2:

F(-1) = 3(-1) + 7 = -3 + 7 = 4

F(0) = 3(0) + 7 = 0 + 7 = 7

F(1) = 3(1) + 7 = 3 + 7 = 10

F(2) = 3(2) + 7 = 6 + 7 = 13

The Range:

Using the numbers 1 to 9 (one time each), the boxes can be filled in the following way to make the equation true. 2:2 = 3×3:9 = 4×4 =16.

An arithmetic contrast among two numbers is known as a percentage. Two sets of provided numbers are considered to be approximately equal with respect to one another in conformity with the rules of proportion if they increase or decrease by the same ratio.

We must create the equation to prove the equality by utilizing each of the numerals 1 through 9 once. Any of the numbers between 1 and 9 are equivalent to 1 and 2.

Let the ratio equal 1.

The comparable ratios are therefore 1:1, 2:2, and 3:3.

2:2 = 3×3:9 = 4×4 =16

Therefore, using the numbers 1 to 9 (one time each), the boxes can be filled in the following way to make the equation true. 2:2 = 3×3:9 = 4×4 =16.

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To determine whether the given vectors in $\mathbb{R}^3$ form a basis for $\mathbb{R}^3$, we need to check if they are linearly independent and span $\mathbb{R}^3$.

To check for linear independence, we set up the augmented matrix:

[

0

5

7

0

0

2

2

0

2

9

11

0

]

⎣

⎡

​

0

0

2

​

5

2

9

​

7

2

11

​

0

0

0

⎦

⎤

We reduce this to echelon form:

[

1

0

2

0

0

1

1

0

0

0

0

0

]

⎣

⎡

​

1

0

0

​

0

1

0

2

1

0

​

0

0

0

​

⎦

⎤

​

Since there is a row of zeros, the rank of the matrix is less than 3, which means the vectors are linearly dependent.

Therefore, the given vectors do not form a basis for $\mathbb{R}^3$.

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Step-by-step explanation:

The formula for the standard error of a proportion is:

SE = sqrt((p*(1-p))/n)

where p is the sample proportion and n is the sample size.

To find p, we need to first calculate the proportion of people who smoke regularly:

p = 71/635 = 0.1118

Now we can plug in the values for p and n:

SE = sqrt((0.1118*(1-0.1118))/635) = 0.0194

Therefore, the standard error for the sample proportion is 0.0194.

blank1 - Word Answer (-5,0)
blank2 - Word Answer (1,0)
blank3 - Word Answer (0,-5)

The x- and y-intercept(s) of an equation are the points where the graph of the equation crosses the x- and y-axis respectively. To find the x-intercept(s), we set y=0 and solve for x. To find the y-intercept, we set x=0 and solve for y.

a. x-intercepts:
0=(x+2)^2-9
9=(x+2)^2
±3=x+2
x=-5 or x=1
So the x-intercepts are (-5,0) and (1,0).

b. y-intercept:
y=(0+2)^2-9
y=4-9
y=-5
So the y-intercept is (0,-5).

Therefore, the correct answers are:
blank1 - Word Answer (-5,0)
blank2 - Word Answer (1,0)
blank3 - Word Answer (0,-5)

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Using synthetic division, we can find that h(-1) = -5.

To find h(-1) using synthetic division, we can use the following steps:

Write the coefficients of the polynomial in descending order of the exponents: 36, 36, 0, 7, 12, 39
Write the value of -1 to the left of the coefficients: -1 | 36 36 0 7 12 39
Bring down the first coefficient: -1 | 36 36 0 7 12 39
         36
Multiply the first coefficient by -1 and write the result under the second coefficient: -1 | 36 36 0 7 12 39
         36 -36
Add the second coefficient and the result: -1 | 36 36 0 7 12 39
         36   0
Repeat steps 4 and 5 for the remaining coefficients: -1 | 36 36 0 7 12 39
         36   0  0 -7 -5
The last number in the bottom row is the remainder, which is the value of h(-1): h(-1) = -5

Therefore, using synthetic division, we can find that h(-1) = -5.

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There are 72 pupils in bertha if the ratio of 1 Bertha to form 1 Florence is 12:20 if there are 120 pupils in Florence.

The problem can be solved by using the concept of ratios and inverse ratios. It involves setting up a proportion based on the given ratio and using algebra to solve for the unknown quantity.

If the ratio of Bertha to Florence is 12:20, then the ratio of Florence to Bertha is 20:12 (inverse ratio).

Let the number of pupils in Bertha be x. Then, the number of pupils in Florence would be:

20/12 * x = 120

Simplifying:

x = 120 * 12/20

x = 72

Therefore, there are 72 pupils in Bertha.

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The equation that shows the volume of the rectangular prism as a product of its edge lengths is (5/3)(1/3)(2/3) = 10/27 in cube. Option 2 is the correct answer.

The quantity of space within a three-dimensional object is its volume. The fundamentals of such forms are described in our page on three-dimensional shapes. Calculating volume is probably not something you will do as frequently as calculating area in the real world.

Even so, it may still be significant. Knowing how to calculate volume will help you determine things like how much room you have for packing when moving house, how much room you need for an office, or how much jam you can put into a jar. It can also be helpful for deciphering what the media means when they discuss a dam's capacity or a river's flow.

The volume of the rectangular prims is given by the formula:

V = lwh

Substituting the values we have:

V = (5/3)(1/3)(2/3) = 10/27

Hence, the equation that shows the volume of the rectangular prism as a product of its edge lengths is (5/3)(1/3)(2/3) = 10/27 in cube. Option 2 is the correct answer.

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The [tex]4\frac{3}{8}-2\frac{1}{8}[/tex] [tex]=2\frac{1}{8}[/tex] This is the correct way to solve the equation it.

The meaning of an equations in algebra is a logical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.

When we say we're "solving" an equation, we truly mean we're "solving" the issue or "answering" the question since, in most circumstances, a formula reflects a problem (or query) of some type. A mathematical phrase with two equal sides and an equal sign is called an equation.

Given,

[tex]4\frac{3}{8}-2\frac{1}{8}[/tex]

[tex]=2\frac{1}{8}[/tex]

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Answer:

Step-by-step explanation:

based on the median of the samples, what is a reasonable estimate of the number of students that bike to school? Round to the nearest whole number based on the median of the samples, what is a reasonable estimate of the number of students that bike to school? Round to the nearest whole number based on the median of the samples, what is a reasonable estimate of the number of students that bike to school? Round to the nearest whole number8957647h3ny4y4y7

it should be +60 not +78

when you multiply 18x by X - 3 you obtain 18x² - 54x . Subtract what you have gotten from 18x²-6x you obtain 60x

the area of the shaded region:

[tex]= 2(the \ area \ of \ a \ quarter \ circle) - (the \ area \ of \ the \ square)\\[/tex]

[tex]= 2\Big(\dfrac{\pi 12^{2} }{4}\Big) - 12^{2} = \dfrac{144 \pi}{2} - 144 = 72 \pi - 144\\[/tex]

[tex]= 72 (\pi - 2) \ m^{2}[/tex]

Answer:

  72π-144 m²

Step-by-step explanation:

You want the area of a shaded region consisting of the overlap of two quarter circles in a 12 m square.

If we draw a diagonal from upper left to lower right through the figure, the shaded area is divided into two 90° segments of a circle of radius 12 m.

The formula for the area of a segment is ...

  A = 1/2r²(θ -sin(θ))

where θ is the measure of the central angle.

For θ = π/2 radians, this is ...

  A = 1/2r²(π/2 -1) . . . . . half the shaded area

Then the whole shaded area is ...

  2 × 1/2r²(π/2 -1) = (12 m)²(π/2 -1) = 72π -144 m²

The area of the shaded region is 72π -144 m².

__

Additional comment

If we expand the shaded area formula, we get ...

  A =1/2πr² -r²

This is recognizable as twice the area of a quarter circle, less the area of the square.

<95141404393>

The solution to the system of linear equations is (47/16, -161/40, 49/8).

To solve the system of linear equations using Gaussian elimination, we first need to write the equations in matrix form.

[2 -3 -5 | -6]
[-6 2 8 | 11]
[3 -1 3 | 5]

Next, we need to use elementary row operations to transform the matrix into reduced row echelon form (RREF). This means that the matrix will have a leading 1 in each row, and all other entries in that column will be 0.

Step 1: Divide the first row by 2 to get a leading 1.

[1 -3/2 -5/2 | -3]
[-6 2 8 | 11]
[3 -1 3 | 5]

Step 2: Add 6 times the first row to the second row to eliminate the -6 in the second row.

[1 -3/2 -5/2 | -3]
[0 5 7 | 7]
[3 -1 3 | 5]

Step 3: Subtract 3 times the first row from the third row to eliminate the 3 in the third row.

[1 -3/2 -5/2 | -3]
[0 5 7 | 7]
[0 7/2 13/2 | 14]

Step 4: Divide the second row by 5 to get a leading 1.

[1 -3/2 -5/2 | -3]
[0 1 7/5 | 7/5]
[0 7/2 13/2 | 14]

Step 5: Add 3/2 times the second row to the first row to eliminate the -3/2 in the first row.

[1 0 -1/2 | -1/2]
[0 1 7/5 | 7/5]
[0 7/2 13/2 | 14]

Step 6: Subtract 7/2 times the second row from the third row to eliminate the 7/2 in the third row.

[1 0 -1/2 | -1/2]
[0 1 7/5 | 7/5]
[0 0 4/5 | 49/10]

Step 7: Divide the third row by 4/5 to get a leading 1.

[1 0 -1/2 | -1/2]
[0 1 7/5 | 7/5]
[0 0 1 | 49/8]

Step 8: Add 1/2 times the third row to the first row to eliminate the -1/2 in the first row.

[1 0 0 | 47/16]
[0 1 7/5 | 7/5]
[0 0 1 | 49/8]

Step 9: Subtract 7/5 times the third row from the second row to eliminate the 7/5 in the second row.

[1 0 0 | 47/16]
[0 1 0 | -161/40]
[0 0 1 | 49/8]

Now the matrix is in RREF, and we can read off the solutions:

x = 47/16
y = -161/40
z = 49/8

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The dot plot allows us to know the exact data we are graphing.

In a dot plot, we have the different possible values on the horizontal axis, and dots above each of these values that count how many times that value has appeared in an experiment.

Instead, in a whisker plot or a box plot we have a representation in a kinda of "rectangle with whiskers".

This type of plot is usefull to find the quartiles of the data set, but not to know exactly how many of each data points we have, like in the dot plot.

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Answer:

Step-by-step explanation:

So all that you have to do is just multiply 15 by 110kg and you'll get a answer of 1500kg total.

Answer:

We can use the given information to find the ant's crawling speed and then use that to calculate how far it can crawl in a given amount of time.

The ant crawls 12 feet in 10 minutes, so its crawling speed is:

12 feet / 10 minutes = 1.2 feet/minute

To find how far the ant can crawl in 20 minutes, we can multiply its crawling speed by the time:

Distance in 20 minutes = crawling speed x time

Distance in 20 minutes = 1.2 feet/minute x 20 minutes

Distance in 20 minutes = 24 feet

Therefore, the ant can crawl 24 feet in 20 minutes.

To find how far the ant can crawl in 30 minutes, we can use the same formula:

Distance in 30 minutes = crawling speed x time

Distance in 30 minutes = 1.2 feet/minute x 30 minutes

Distance in 30 minutes = 36 feet

Therefore, the ant can crawl 36 feet in 30 minutes.

Answer:

20 mins=24 feet

30 mins=36 feet

Step-by-step explanation:

if 12 feet = 10 mins and 10 times 2 = 20 then 12 times 2 =24

same for 30 mins.

10 times 3= 30 then 12 times 3= 36

Step-by-step explanation:

Refer to pic.............

These are the Critical numbers of the function within the given interval.
Therefore, the answer is \(x=\frac{\pi }{6}, \frac{5 \pi }{6}\).

The Critical numbers of a function are the values of x that make the derivative of the function equal to zero. To find the eritical numbers of the given function, we need to find the derivative of the function and set it equal to zero.

The derivative of the function \(f(x)=-x+\sin (2 x)\) is \(f'(x)=-1+2\cos (2 x)\).

Setting the derivative equal to zero, we get:

\(-1+2\cos (2 x)=0\)

\(\cos (2 x)=\frac{1}{2}\)

Using the inverse cosine function, we get:

\(2 x=\cos ^{-1}\left(\frac{1}{2}\right)\)

\(2 x=\frac{\pi }{3}\) or \(2 x=\frac{5 \pi }{3}\)

Dividing both sides by 2, we get:

\(x=\frac{\pi }{6}\) or \(x=\frac{5 \pi }{6}\)

These are the Critical numbers of the function within the given interval.

Therefore, the answer is \(x=\frac{\pi }{6}, \frac{5 \pi }{6}\).

Learn more about  Critical numbers

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1) The ratios 1/3 , 7/21 are in proportion.

2) The ratios 2/5 , 40/16 are not in proportion.

3) the ratios 48/9 , 16/3 are in proportion.

We know that two ratios are said to be in proportion when both the ratios are equivalent.

Consider 1/3 , 7/21

Here 7/21 = (7 × 1)/(7 × 3)

                = 1/3

This means 1/3 = 7/21

Thus, the ratios 1/3 , 7/21 are in proportion.

Consider 48/9, 16/3

Here 48/9 = (16 × 3)/(3 × 3)

                = 16/3

This means 48/9 = 16/3

Thus, the ratios 48/9, 16/3 are in proportion.

Now consider 2/5 , 40/16

Here 40/16 = (8 × 5)/(8 × 2)

                = 5/2

This means 2/5 ≠ 5/2

Thus, the ratios 2/5 , 40/16 are not in proportion.

Learn more about proportion here:

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