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The table gives the volume of cylinders where the height is a function of the radius of the base. Given the radius, the height can be modeled with the function [tex]\( h(r) \)[/tex], and the volume of the cylinder with the function [tex]\( V(h) \)[/tex].

[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
$r$ & $h(r)$ & & $V(h)$ \\
\hline
2 & $4 \pi$ & & $16 \pi$ \\
\hline
8 & $64 \pi$ & & $1,024 \pi$ \\
\hline
10 & $10 / 2 \pi$ & & $2,000 \pi$ \\
\hline
12 & $144 \pi$ & & $3,456 \pi$ \\
\hline
\end{tabular}
\][/tex]

Which values complete the table for [tex]\( h(r) \)[/tex]?
[tex]\[
\begin{tabular}{|c|}
\hline
$h(r)$ \\
\hline
$4 \pi$ \\
\hline
$16 \pi$ \\
\hline
$20 \pi$ \\
\hline
$24 \pi$ \\
\hline
\end{tabular}
\][/tex]

[tex]\[
\begin{tabular}{|c|}
\hline
$h(r)$ (cm) \\
\hline
4 \\
\hline
16 \\
\hline
20 \\
\hline
24 \\
\hline
\end{tabular}
\][/tex]

Sagot :

GinnyAnsweridn
Alright, let's analyze the problem step-by-step.

Given:
1. The table provides the volume of cylinders where the height is a function of the radius of the base.
2. The formula for the volume [tex]\( V \)[/tex] of a cylinder is [tex]\( V = \pi r^2 h \)[/tex].

We can use this formula to find the height [tex]\( h \)[/tex] when we know the volume [tex]\( V \)[/tex] and the radius [tex]\( r \)[/tex].

Given the table:

[tex]\[ \begin{array}{|c|c|c|c|} \hline r & \text{Radius (cm)} & \text{Height (cm)} & \text{Volume} \, (V = \pi r^2 h) \\ \hline 2 & 2 & ? & 16\pi \\ \hline 8 & 8 & ? & 1,024\pi \\ \hline 10 & 10 & ? & 2,000\pi \\ \hline 12 & 12 & ? & 3,456 \pi \\ \hline \end{array} \][/tex]

We are supposed to find the values of [tex]\( h \)[/tex] that complete the table:

1. For [tex]\( r = 2 \)[/tex] and [tex]\( V = 16\pi \)[/tex]:

[tex]\[ 16\pi = \pi (2)^2 h \\ 16\pi = 4\pi h \\ \Rightarrow h = \frac{16\pi}{4\pi} = 4 \][/tex]

So, the height when the radius is 2 cm is [tex]\( 4 \)[/tex] cm.

2. For [tex]\( r = 8 \)[/tex] and [tex]\( V = 1,024\pi \)[/tex]:

[tex]\[ 1,024\pi = \pi (8)^2 h \\ 1,024\pi = 64\pi h \\ \Rightarrow h = \frac{1,024\pi}{64\pi} = 16 \][/tex]

So, the height when the radius is 8 cm is [tex]\( 16 \)[/tex] cm.

3. For [tex]\( r = 10 \)[/tex] and [tex]\( V = 2,000\pi \)[/tex]:

[tex]\[ 2,000\pi = \pi (10)^2 h \\ 2,000\pi = 100\pi h \\ \Rightarrow h = \frac{2,000\pi}{100\pi} = 20 \][/tex]

So, the height when the radius is 10 cm is [tex]\( 20 \)[/tex] cm.

4. For [tex]\( r = 12 \)[/tex] and [tex]\( V = 3,456\pi \)[/tex]:

[tex]\[ 3,456\pi = \pi (12)^2 h \\ 3,456\pi = 144\pi h \\ \Rightarrow h = \frac{3,456\pi}{144\pi} = 24 \][/tex]

So, the height when the radius is 12 cm is [tex]\( 24 \)[/tex] cm.

Completing the table with the values of [tex]\( h \)[/tex]:

[tex]\[ \begin{array}{|c|c|c|c|} \hline r & \text{Radius (cm)} & h \, (\text{Height (cm)}) & V \, (\text{Volume}) \\ \hline 2 & 2 & 4 & 16\pi \\ \hline 8 & 8 & 16 & 1,024\pi \\ \hline 10 & 10 & 20 & 2,000\pi \\ \hline 12 & 12 & 24 & 3,456\pi \\ \hline \end{array} \][/tex]
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