Question

the table shows the volume of two similar solids, right circular cylinder a and right circular cylinder b. the radius of right circular cylinder a is 2 units. the surface area of right circular cylinder a is kπ square units, and the surface area of right circular cylinder b is nπ square units , where k and n are constants. what is the value of n - k?(the surface area of a right circular cylinder with radius r and height h is 2πr² + 2πrh)

Explanation:

Step1: Find height of cylinder A

Given $V_A = 32\pi=\pi r^2h_A$, $r = 2$, so $32\pi=\pi\times2^2\times h_A$, $h_A = 8$.

Step2: Calculate surface - area of cylinder A

$k\pi=2\pi r^2+2\pi rh_A=2\pi\times2^2 + 2\pi\times2\times8=40\pi$, so $k = 40$.

Step3: Find scale - factor of volumes

$\frac{V_B}{V_A}=\frac{864\pi}{32\pi}=27$, scale - factor of lengths $s=\sqrt[3]{27}=3$.

Step4: Calculate surface - area of cylinder B

Surface - area ratio is $s^2 = 9$. So $n\pi=k\pi\times9 = 360\pi$, $n = 360$.

Step5: Calculate $n - k$

$n - k=360 - 40=320$.

Answer:

320