Answer :
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]
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]
