Question 346·Hard·Math

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Sphere P has a radius of $x$ units. Sphere Q has a surface area that is $144\pi$ square units greater than the surface area of sphere P. The function $f$ gives the volume of sphere Q, in cubic units, as a function of $x$ for $x>0$ . Which of the following defines $f$ ?

A sphere’s surface area uses the radius squared, while its volume uses the radius cubed. If one sphere has a fixed amount more surface area, write an equation using $4\pi r^2$ and solve for its radius before using $V=\frac43\pi r^3$ . Extra area adds to the squared radius, not directly to the radius.

Hints

Hint 1
Call sphere Q’s radius $r$ . A sphere’s surface area is $4\pi r^2$ . How can you write Q’s surface area as P’s surface area plus the given increase?
Hint 2
Both surface areas contain $4\pi$ . Divide the whole equation by that factor. What does the given $144\pi$ add to $r^2$ ?
Hint 3
A radius is a length, so take the positive square root. Then put that radius into the sphere’s volume formula, $V=\frac43\pi r^3$ . What happens when you cube a square root?

Step-by-step

Find Q’s radius, then its volume
Step 1
Write the surface-area equation
Let $r$ be sphere Q’s radius. A sphere’s surface area is $4\pi$ times its radius squared. Q’s area is $144\pi$ greater than P’s, so:
$4\pi r^2=4\pi x^2+144\pi.$
Step 2
See which radius the equation describes
Type $4\pi y^2=4\pi x^2+144\pi$ in Desmos, using $y$ for Q’s radius $r$ . Desmos draws two branches. For the given $x>0$ , the upper branch has a positive $y$ ; the lower branch cannot represent a radius.
Step 3
Find Q’s squared radius
Divide the surface-area equation by $4\pi$ :
$r^2=x^2+\frac{144\pi}{4\pi}.$
Type $\frac{144\pi}{4\pi}$ on the next Desmos line. It displays $36$ . Replace the quotient:
$r^2=x^2+36.$
The $36$ belongs with $x^2$ , not with $x$ .
Step 4
Find Q’s radius
Take the positive square root, because a radius is a length:
$r=\sqrt{x^2+36}.$
Step 5
Cube Q’s radius
Cube Q’s radius to prepare for the sphere’s volume formula:
$r^3=\left(\sqrt{x^2+36}\right)^3.$
Step 6
Rewrite the cubed radius
A square root is a $1/2$ power, so cubing it gives a $3/2$ power. Rewrite the cube of Q’s radius:
$r^3=(x^2+36)^{3/2}.$
Step 7
Write Q’s volume function
A sphere’s volume, the space inside it, is $V=\frac43\pi r^3$ . Substitute Q’s cubed radius:
$f(x)=\frac43\pi(x^2+36)^{3/2}.$
So this function gives sphere Q’s volume in cubic units. Choice B.

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SAT Geometry and TrigonometryIntermediateCoreFind surface area and volume

Question 346·Hard·Math

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