Let's solve this step-by-step:
(a) Work out the length of cylinder Q.
Given:
- Surface area of cylinder P, [tex]\( S_P = 90 \)[/tex] cm²
- Surface area of cylinder Q, [tex]\( S_Q = 810\pi \)[/tex] cm²
- Length of cylinder P, [tex]\( L_P = 4 \)[/tex] cm
We want to find the length of cylinder Q, [tex]\( L_Q \)[/tex].
Since the cylinders are similar, the ratio of the surface areas is equal to the square of the scale factor (k^2), where k is the scale factor between the dimensions of the cylinders. Let's calculate k:
[tex]\[
k^2 = \frac{S_Q}{S_P}
\][/tex]
Plug in the values:
[tex]\[
k^2 = \frac{810\pi}{90} = 9\pi
\][/tex]
Find the scale factor, k, by taking the square root:
[tex]\[
k = \sqrt{9\pi} = 3\sqrt{\pi}
\][/tex]
Now, because the cylinders are similar, the length of cylinder Q is k times the length of cylinder P:
[tex]\[
L_Q = L_P \cdot k
\][/tex]
Plug in the values:
[tex]\[
L_Q = 4 \cdot 3\sqrt{\pi} = 12\sqrt{\pi} \text{ cm}
\][/tex]
(b) Work out the volume of cylinder Q.
Given:
- Volume of cylinder P, [tex]\( V_P = 100\pi \)[/tex] cm³
We want to find the volume of cylinder Q, [tex]\( V_Q \)[/tex].
The volume scale factor is the cube of the dimension scale factor (since volume is a measure in cubic units), so:
[tex]\[
V_Q = V_P \cdot k^3
\][/tex]
We already know that [tex]\( k = 3\sqrt{\pi} \)[/tex], now let's cube it to find [tex]\( k^3 \)[/tex]:
[tex]\[
k^3 = (3\sqrt{\pi})^3 = 3^3 \cdot (\sqrt{\pi})^3 = 27\pi\sqrt{\pi}
\][/tex]
Now calculate [tex]\( V_Q \)[/tex]:
[tex]\[
V_Q = 100\pi \cdot 27\pi\sqrt{\pi} = 2700\pi^2\sqrt{\pi}
\][/tex]
The question asks for the answer as a multiple of [tex]\( \pi \)[/tex]. We can express [tex]\( \pi^2\sqrt{\pi} \)[/tex] as [tex]\( \pi^{2.5} \)[/tex] (since [tex]\( \sqrt{\pi} = \pi^{0.5} \)[/tex]):
[tex]\[
V_Q = 2700\pi^{2.5}
\][/tex]
To express [tex]\( V_Q \)[/tex] as a multiple of [tex]\( \pi \)[/tex], we take out the factor of [tex]\( \pi \)[/tex] explicitly:
[tex]\[
V_Q = 2700\pi^2\sqrt{\pi} = (2700\sqrt{\pi})\pi
\][/tex]
So, the volume of cylinder Q is [tex]\( 2700\sqrt{\pi} \)[/tex] times [tex]\( \pi \)[/tex] (as a multiple of [tex]\( \pi \)[/tex]).
