Question 152·Medium·Math

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A right circular cylinder has a total surface area of $96\pi$ square centimeters. If the height of the cylinder is twice its radius, what is the volume, in cubic centimeters, of the cylinder?

For cylinder problems where you are given surface area and a relationship between dimensions but asked for volume, first write the general surface area formula, substitute the given relationship (such as $h = 2r$ ) so that you have a single variable, and solve that equation carefully. Then, use the found dimension(s) in the volume formula—double-check that you are using the correct relationship (like $h = 2r$ ) and that your final units are cubic, not square, to avoid mixing up surface area and volume.

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Recall the two key formulas

Write down the formulas for the total surface area of a cylinder, $S = 2\pi r^2 + 2\pi rh$ , and for the volume, $V = \pi r^2 h$ .

Use the relationship between height and radius

Substitute $h = 2r$ into the surface area formula so that the equation has only one variable, $r$ .

Solve step by step, then switch formulas

After you find $r^2$ (and then $r$ ), use $h = 2r$ to find $h$ . Then plug $r$ and $h$ into the volume formula, not back into the surface area formula.

Check units and what is being asked

Make sure your final calculation gives cubic centimeters (volume), not square centimeters (surface area).

Desmos Guide

Model the surface area equation in Desmos

In Desmos, enter the expression A(r) = 2*pi*r^2 + 2*pi*r*(2*r) to represent the total surface area in terms of $r$ .

Find the radius that gives the correct surface area

Add a second expression y = 96*pi. Then, graph y = A(r) by entering y = 2*pi*r^2 + 2*pi*r*(2*r) and look for the intersection point of this graph with the horizontal line y = 96*pi. The $r$ -coordinate of this intersection is the radius.

Compute the corresponding height

Once you know the value of $r$ from the graph, use the relationship $h = 2r$ . You can type something like h = 2*r_value (replacing r_value with the radius you found) to let Desmos calculate the height.

Use Desmos to calculate the volume

Finally, enter V = pi*(r_value^2)*h_value, using your radius and height numbers, to have Desmos output the volume. The numerical result is the volume in cubic centimeters.

Step-by-step Explanation

Write the formulas and use the height–radius relationship

For a right circular cylinder:

Total surface area: $S = 2\pi r^2 + 2\pi rh$ (two circular bases plus the side)
Volume: $V = \pi r^2 h$

We are told the total surface area is $96\pi$ and the height is twice the radius, so $h = 2r$ .

Set up and simplify the surface area equation

Substitute $h = 2r$ into the surface area formula and set it equal to $96\pi$ :

\begin{aligned} 2\pi r^2 + 2\pi r(2r) &= 96\pi \\ 2\pi r^2 + 4\pi r^2 &= 96\pi \\ 6\pi r^2 &= 96\pi \end{aligned}

Now divide both sides by $6\pi$ to solve for $r^2$ :

$6\pi r^2 = 96\pi \Rightarrow r^2 = 16$ .

Find the radius and the height

From $r^2 = 16$ we get $r = 4$ (radii are positive).

Since $h = 2r$ , the height is

$h = 2(4) = 8$ .

Compute the volume using $r$ and $h$

Now use the volume formula $V = \pi r^2 h$ with $r = 4$ and $h = 8$ :

\begin{aligned} V &= \pi (4)^2 (8) \\ &= \pi (16)(8) \\ &= 128\pi \end{aligned}

So the volume of the cylinder is $128\pi$ cubic centimeters, which corresponds to choice C.

Question 152·Medium·Math

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation