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