Question 634·Hard·Math

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A right circular cylinder has a volume of $1,800\pi$ cubic centimeters. The cylinder’s height $h$ , in centimeters, is 5 centimeters less than twice its radius $r$ , in centimeters. Which equation represents this relationship?

For word problems that ask you to "write an equation" about a geometric shape, first recall the relevant formula (here, cylinder volume), then rewrite any verbal relationships (like "5 less than twice the radius") as algebraic expressions with clear attention to order and minus signs. Substitute these expressions into the formula, simplify step by step, and only at the end move all terms to one side to match the SAT’s standard "0 = ..." format; then compare carefully with the answer choices, checking especially the signs of each term.

Hints

Hint 1
Use the cylinder volume formula
What is the standard formula for the volume $V$ of a right circular cylinder in terms of its radius $r$ and height $h$ ? Set this equal to $1{,}800\pi$ .
Hint 2
Carefully interpret the height description
"5 centimeters less than twice its radius" describes $h$ in terms of $r$ . If "twice its radius" is $2r$ , how do you write "5 less than" that quantity?
Hint 3
Form a single equation in r
Substitute your expression for $h$ into the volume formula so that the only variable is $r$ . Then simplify the left side and move everything to one side so the equation equals 0.
Hint 4
Check signs when moving terms
After you have something like $(\text{expression in } r) = 1{,}800$ , think about what happens to the $1{,}800$ when you move it to the other side. Does it become $+1{,}800$ or $-1{,}800$ on the side with 0?

Desmos guide

Step 1
Model the volume equation in Desmos
In Desmos, use $x$ instead of $r$ . Enter the expression for the left side of the simplified volume equation as a function: type y = x^2(2x - 5).
Step 2
Represent the given volume
Enter a second function for the constant volume: type y = 1800. The intersection points of these two graphs show values of $x$ that satisfy $x^2(2x - 5) = 1800$ .
Step 3
Connect the graph to the answer choices
From the graph, you know the underlying equation before rearranging is $x^2(2x - 5) = 1800$ . If you expand $x^2(2x - 5)$ and then move $1800$ to the same side as the $x$ terms so that the other side is 0, you will get a cubic expression. Choose the answer option whose equation matches that expanded and rearranged form (with $r$ in place of $x$ ).

Step-by-step

Step 1
Write the volume formula and plug in the given volume
For a right circular cylinder, the volume formula is $V = \pi r^2 h$ .

We are told the volume is $1{,}800\pi$ cubic centimeters, so set up:
$\pi r^2 h = 1{,}800\pi$
2
Step 2
Translate the height relationship into an equation
The problem says: "The cylinder’s height $h$ is 5 centimeters less than twice its radius $r$ ."

"Twice its radius" means $2r$ .

"5 centimeters less than" that means you subtract 5 from $2r$ .

So the relationship is: $h = 2r - 5$ (not $2r + 5$ ).
Step 3
Substitute for h and simplify the equation
Replace $h$ in the volume equation with $2r - 5$ :
$\pi r^2(2r - 5) = 1{,}800\pi$
Now divide both sides by $\pi$ (since $\pi \ne 0$ ):
$r^2(2r - 5) = 1{,}800$
Next, expand the left side:
$r^2(2r - 5) = 2r^3 - 5r^2$
So you have:
$2r^3 - 5r^2 = 1{,}800$
Step 4
Move all terms to one side and match an answer choice
To get an equation equal to 0, subtract $1{,}800$ from both sides:
$2r^3 - 5r^2 - 1{,}800 = 0$
This is equivalent to
$0 = 2r^3 - 5r^2 - 1{,}800$
which matches choice D.

Lessons that teach this

SAT Geometry and TrigonometryIntermediateCoreFind surface area and volume

Question 634·Hard·Math

Strategy

Hints

Desmos guide

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Desmos guide

Step-by-step

Lessons that teach this