Answer:
12 cm
Step-by-step explanation:
Since the cylinders are mathematically similar, their surface areas are proportional to the square of their linear dimensions.
Let's denote the length of cylinder Q as [tex] l_q [/tex].
For cylinder P:
Total surface area:
[tex] A_p = 2\pi r_p^2 + 2\pi r_p h_p [/tex]
Given:
[tex] A_p = 90 [/tex] cm², [tex] h_p = 4 [/tex] cm
For cylinder Q:
Total surface area
[tex] A_q = 2\pi r_q^2 + 2\pi r_q h_q [/tex]
Given:
[tex] A_q = 810 [/tex] cm²
We have the following ratio of surface areas:
[tex] \dfrac{A_q}{A_p} = \left(\dfrac{l_q}{4}\right)^2 [/tex]
[tex] \dfrac{810}{90} = \left(\dfrac{l_q}{4}\right)^2 [/tex]
[tex] 9 = \left(\dfrac{l_q}{4}\right)^2 [/tex]
Now, solve for [tex] l_q [/tex]:
[tex] \left(\dfrac{l_q}{4}\right)^2 = 9 [/tex]
[tex] \dfrac{l_q^2}{16} = 9 [/tex]
[tex] l_q^2 = 9 \times 16 [/tex]
[tex] l_q^2 = 144 [/tex]
[tex] l_q = \sqrt{144} [/tex]
[tex] l_q = 12 [/tex]
So, the length of cylinder Q is 12 cm.
