Question

describing composite functions
the volume, in cubic feet, of a right cylindrical silo of height ( h ) and radius ( r ) is ( v = pi r^2 h ). the height of the silo is ( h(r) = 3.5r ).
which statements are true regarding the functions described? choose three correct answers.

• the output of ( v ) is the input of ( h ).
• to get a volume of 100 cubic feet, the radius must be 2 feet.
• the domain of ( v(h(r)) ) is restricted to values of ( r ) greater than 0.
• ( v(h(r)) = 3.5pi r^3 )

Explanation:

Step1: Analyze composite function input/output

The function $h(r)$ outputs height, which is the input for $V$. The output of $V$ is volume, which is not the input of $h$, so this statement is false.

Step2: Verify volume for $r=2$

First, find $V(h(r))$. Substitute $h(r)=3.5r$ into $V=\pi r^2 h$:
$V(h(r))=\pi r^2(3.5r)=3.5\pi r^3$
Now plug $r=2$:
$V(h(2))=3.5\pi (2)^3=3.5\pi \times8=28\pi\approx87.96$ cubic feet, which is not 100. So this statement is false.

Step3: Check domain of $V(h(r))$

$r$ represents the radius of a silo, so it must be a positive real number (cannot be 0 or negative). Thus the domain of $V(h(r))$ is $r>0$, so this statement is true.

Step4: Derive the composite function

Substitute $h(r)=3.5r$ into $V=\pi r^2 h$:
$V(h(r))=\pi r^2(3.5r)=3.5\pi r^3$
This matches the given expression, so this statement is true.
Wait, correction: Recheck the problem says "choose three correct answers"—wait, re-express $V$ properly: the $r$ in $V$ is the same radius as in $h(r)$. So $V(h(r)) = \pi r^2 \times 3.5r = 3.5\pi r^3$, correct. Domain: $r>0$ is true. Wait, did I miss one? Wait no, recheck the first statement: no, input of $h$ is $r$, output is $h$. Input of $V$ is $r$ and $h$. The composite uses output of $h$ as input (height) for $V$, not the other way. The second statement: $3.5\pi r^3=100$ → $r^3=\frac{100}{3.5\pi}\approx9.09$ → $r\approx2.09$, not 2, so false. Wait, maybe I misread: wait the problem says "choose three"—wait no, let's re-express:
Wait, $V(h(r)) = \pi r^2 h(r) = \pi r^2 (3.5r) = 3.5\pi r^3$, that is correct (fourth statement true). Domain: radius can't be non-positive, so third statement true. Wait, is there a third correct? Wait no, wait maybe I messed up the first statement: no, output of $V$ is volume, input of $h$ is radius, so that's wrong. Second is wrong. Wait, maybe the problem's $V$ is $V(h)$? No, the problem says $V=\pi r^2 h$, height is $h(r)=3.5r$. Oh wait, no—wait, maybe I miswrote the composite: $V$ is a function of $r$ and $h$, but when we make it a function of $r$ only, we substitute $h=3.5r$, so $V(r) = 3.5\pi r^3$, which is $V(h(r))$. So the correct statements are:

1. The domain of $V(h(r))$ is restricted to values of $r$ greater than 0.
2. $V(h(r))=3.5\pi r^3$

Wait, but the problem says "choose three"—wait, no, let's recalculate the second statement: if $V=100$, then $3.5\pi r^3=100$ → $r^3=\frac{100}{3.5\times3.1416}\approx\frac{100}{11}\approx9.09$ → $r\approx2.09$, which is not 2, so that's false. The first statement is false. So maybe the problem has a typo? No, wait, no—wait, maybe $V$ is $V(h)$, where $h$ is height, and $r$ is a function of $h$? No, the problem says $h(r)=3.5r$, so height is a function of radius. So only two correct? No, wait, no—wait, I think I made a mistake: the first statement: "The output of $V$ is the input of $h$"—no, input of $h$ is $r$, output of $V$ is volume. That's wrong. Wait, maybe the other way: output of $h$ is input of $V$, which is true, but that's not the statement. So the two correct are the third and fourth. But the problem says "choose three"—wait, no, wait let's recheck the composite function again:
Wait $V = \pi r^2 h$, $h(r)=3.5r$. So $V(h(r)) = \pi r^2 (3.5r) = 3.5\pi r^3$, correct. Domain: $r>0$, correct. Is there a third? Wait, maybe the second statement: if $r=2$, $V=3.5\pi(8)=28\pi\approx87.96$, not 100. So that's wrong. The first is wrong. So maybe the problem has a mistake, but based on calculations, the correct statements are:

Answer:

• The domain of $V(h(r))$ is restricted to values of $r$ greater than 0.
• $V(h(r)) = 3.5\pi r^3$

Wait, but the problem says "choose three"—wait, no, I must have misread. Wait, is $V$ a function of $h$? Like $V(h)=\pi (\frac{h}{3.5})^2 h$? No, the problem says $h(r)=3.5r$, so $r=\frac{h}{3.5}$. But the problem says $V=\pi r^2 h$, so $V(h)=\pi (\frac{h}{3.5})^2 h = \frac{\pi h^3}{12.25}$, but that's not what's asked. The question is about the given statements. So the only two true statements are the third and fourth. But if we must choose three, maybe I made a mistake in the second statement: $3.5\pi r^3=100$ → $r=\sqrt[3]{\frac{100}{3.5\pi}}\approx\sqrt[3]{9.09}\approx2.09$, which is approximately 2, but not exactly. So it's not true. The first statement is false. So the correct answers are the third and fourth options.