The circumference of a great circle (a circle formed by slicing the sphere through its center) of sphere is , where . The surface area of sphere is greater than the surface area of sphere . The volume of sphere is how many times the volume of sphere ?

Solution available with step-by-step explanation

Super-Hard SAT Math Question 113 | Free SAT Prep | Aniko

00:00

Question 113·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

0 / 200

The circumference of a great circle (a circle formed by slicing the sphere through its center) of sphere $A$ is $12\pi x$ , where $x>0$ . The surface area of sphere $B$ is $756\pi x^2$ greater than the surface area of sphere $A$ . The volume of sphere $B$ is how many times the volume of sphere $A$ ?

For sphere comparison problems, aim to find a radius ratio. Convert each given measurement into radius using the matching formula (great-circle circumference $C=2\pi r$ , surface area $S=4\pi r^2$ ). If you’re given a relationship between areas (like a difference), compute one area first, then use the relationship to get the other. Finally, use that volume scales as the cube of the radius ratio: $\frac{V_B}{V_A}=\left(\frac{r_B}{r_A}\right)^3$ .

Hints

Click me!

Start with the great-circle circumference

Use $C=2\pi r$ to find $r_A$ from $12\pi x$ .

Turn $r_A$ into a surface area

Compute $S_A$ using $S=4\pi r^2$ .

Apply the “greater than” statement carefully

Translate “ $756\pi x^2$ greater than” into an equation for $S_B$ in terms of $S_A$ .

Convert $S_B$ back to a radius

Solve $4\pi r_B^2=S_B$ for $r_B$ (and keep the positive root).

Use the scaling rule for volumes

Once you have $\frac{r_B}{r_A}$ , compute $\left(\frac{r_B}{r_A}\right)^3$ .

Desmos Guide

Compute $r_A$ and $S_A$ (set $x=1$ )

Because $x$ cancels in the ratio, set $x=1$ .

Enter in Desmos:

rA=(12*pi*1)/(2*pi)
SA=4*pi*rA^2

Add the surface-area difference to get $S_B$

Enter:

SB=SA+756*pi*1^2

Solve for $r_B$ and cube the radius ratio

Enter:

rB=sqrt(SB/(4*pi))
(rB/rA)^3

This value is how many times $V_B$ is compared to $V_A$ (only reveal the final choice after computing).

Step-by-step Explanation

Find the radius of sphere $A$ from its great-circle circumference

A great circle has the same radius as the sphere, and its circumference is $C=2\pi r$ .

r_A=\frac{12\pi x}{2\pi}=6x.

Compute the surface area of sphere $A$

Surface area is $S=4\pi r^2$ , so

S_A=4\pi(6x)^2=4\pi\cdot 36x^2=144\pi x^2.

Use the surface-area difference to find $S_B$

The problem says sphere $B$ has surface area $756\pi x^2$ greater than sphere $A$ :

S_B=S_A+756\pi x^2=144\pi x^2+756\pi x^2=900\pi x^2.

Convert $S_B$ to the radius $r_B$

Using $S=4\pi r^2$ again,

4\pi r_B^2=900\pi x^2 \Rightarrow r_B^2=\frac{900\pi x^2}{4\pi}=225x^2.

Since $x>0$ , the radius is

r_B=15x.

Cube the radius ratio to get the volume ratio

Volume scales with $r^3$ , so

\frac{V_B}{V_A}=\left(\frac{r_B}{r_A}\right)^3=\left(\frac{15x}{6x}\right)^3=\left(\frac{5}{2}\right)^3=\frac{125}{8}.

Therefore, the volume of sphere $B$ is $\frac{125}{8}$ times the volume of sphere $A$ .

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation