A right circular cylinder has a volume of cubic units and a total surface area (including both circular bases) of square units. If the height of the cylinder is less than its diameter, which choice is the height of the cylinder?

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Super-Hard SAT Math Question 46 | Free SAT Prep | Aniko

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Question 46·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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A right circular cylinder has a volume of $100\pi$ cubic units and a total surface area (including both circular bases) of $90\pi$ square units. If the height of the cylinder is less than its diameter, which choice is the height of the cylinder?

For cylinder area/volume problems with unknown $r$ and $h$ , write both formulas and eliminate one variable quickly (usually solve for $h$ from volume and substitute into surface area). If the resulting equation can have more than one positive solution, use any extra condition (like comparing height and diameter) to select the valid one rather than assuming the first solution you find is unique.

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Start with the two cylinder formulas

Use $V=\pi r^2h$ and total surface area $S=2\pi r^2+2\pi rh$ (this includes both bases).

Get $h$ alone from the volume equation

Solve $\pi r^2h=100\pi$ for $h$ in terms of $r$ , then substitute into the surface area equation.

Use the extra condition at the end

After you find possible values, check which one makes the height less than the diameter $2r$ .

Desmos Guide

Graph the cubic for the radius

Enter the function $y=x^3-45x+100$ .

Find the positive zeros

Find the $x$ -intercepts of the graph (the positive ones are possible radii $r$ ).

Compute the height from each radius

For each positive intercept $r$ , compute $h=100/r^2$ (you can type $100/x^2$ and evaluate it at each intercept’s $x$ -value).

Apply the condition $h<2r$

Check which computed height is less than $2r$ (twice the radius). The height that satisfies this condition matches one of the answer choices.

Step-by-step Explanation

Write equations for volume and surface area

Volume:

\pi r^2h=100\pi \Rightarrow r^2h=100

Total surface area (including both bases):

2\pi r^2+2\pi rh=90\pi \Rightarrow r^2+rh=45

Eliminate $h$ to solve for $r$

From $r^2h=100$ , solve for $h$ :

h=\frac{100}{r^2}

Substitute into $r^2+rh=45$ :

r^2+r\left(\frac{100}{r^2}\right)=45 \Rightarrow r^2+\frac{100}{r}=45

Multiply both sides by $r$ :

r^3+100=45r \Rightarrow r^3-45r+100=0

Find a positive root and compute the corresponding height

Test a reasonable integer value for $r$ (a factor of 100). Try $r=5$ :

5^3-45(5)+100=125-225+100=0

So $r=5$ is a solution. Then

h=\frac{100}{r^2}=\frac{100}{25}=4

Use the condition to confirm the correct height

The diameter is $2r=10$ . The condition says $h<2r$ , and for $r=5$ we got $h=4$ , which satisfies $4<10$ .

Therefore, the height of the cylinder is 4.

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