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

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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 $A$ ?

A scale factor tells you how much a length changes from one sphere to another. The great circle’s circumference gives the first sphere’s radius; then the surface-area difference lets you find the second sphere’s total surface area. Use Desmos for the area arithmetic. The surface-area ratio is the square of the radius factor, but the requested volume ratio is its cube.

Hints

Hint 1
The great circle passes through the sphere’s center, so its radius is also the sphere’s radius. Circumference is $2\pi r$ . What radius gives the stated circumference?
Hint 2
Sphere surface area is $4\pi r^2$ . Find sphere A’s area first, then add the stated difference to get sphere B’s total area.
Hint 3
A linear scale factor compares corresponding lengths, such as the two radii. The surface-area ratio is its square. After finding that factor, what power gives the volume ratio?

Step-by-step

Find the radius factor, then cube it
Step 1
Find sphere A’s radius
A great circle passes through the sphere’s center, so its radius is the sphere’s radius, $r_A$ . Its circumference is $2\pi r_A$ , so the given circumference gives: $12\pi x=2\pi r_A$ . Divide by $2\pi$ : $r_A=6x$ . This is a radius, not a diameter.
Step 2
Find sphere A’s surface area
A sphere’s surface area is $4\pi r^2$ . Substitute $r_A=6x$ , squaring both $6$ and $x$ : $S_A=4\pi(6x)^2=4(6)^2\pi x^2$ . Type $4(6)^2$ in Desmos. It shows $144$ , so sphere A’s area is $144\pi x^2$ .
Step 3
Find sphere B’s total surface area
Sphere B’s area is greater by $756\pi x^2$ , so add that amount to A’s area: $S_B=(144+756)\pi x^2$ . Add $144+756$ in Desmos. It shows $900$ , giving $S_B=900\pi x^2$ . The added amount alone is not B’s total area.
Step 4
Turn the area ratio into a radius factor
Let $k$ be the radius factor from A to B: $r_B=kr_A$ . Because sphere surface area uses $r^2$ , multiplying a radius by $k$ multiplies its area by $k^2$ : $k^2=\frac{S_B}{S_A}=\frac{900}{144}$ . Type $900/144$ in Desmos. It shows $6.25$ ; its fraction button shows $\frac{25}{4}$ . So $k^2=\frac{25}{4}$ .
Step 5
Recover the radius factor
Take the positive square root of $k^2=\frac{25}{4}$ , since a ratio of radii must be positive:
$k=\sqrt{\frac{25}{4}}$
$k=\frac52$ . This is a length factor, not yet the volume factor.
Step 6
Find the volume ratio
Sphere volume uses $r^3$ , so a radius factor of $k$ gives a volume factor of $k^3$ . Square the radius factor for surface area, but cube it for volume. Type $\left(\frac52\right)^3$ in Desmos. It shows $15.625$ ; its fraction button shows $\frac{125}{8}$ . Sphere B’s volume is $\frac{125}{8}$ times sphere A’s. Choice A.

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

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