Question

cylinder a and cylinder b are each right circular cylinders with a radius of 4 inches. the height of cylinder a is half the height of cylinder b, and the total surface area of cylinder a is 168π square inches. if the cylinders are glued together along a circular base, what is the height, in inches, of the resulting cylinder? a 8 b 17 c 34 d 51

Explanation:

Step1: Write the formula for the total surface area of a cylinder

The total surface area formula of a right - circular cylinder is \(S = 2\pi r^{2}+2\pi rh\). Given \(r = 4\) inches, so \(S=2\pi\times4^{2}+2\pi\times4h=32\pi + 8\pi h\).

Step2: Solve for the height of cylinder A

Since the total surface area of cylinder A is \(168\pi\), we set up the equation \(32\pi+8\pi h=168\pi\).
Subtract \(32\pi\) from both sides: \(8\pi h=168\pi - 32\pi=136\pi\).
Divide both sides by \(8\pi\): \(h=\frac{136\pi}{8\pi}=17\) inches. This is the height of cylinder A.

Step3: Find the height of cylinder B

Since the height of cylinder A is half the height of cylinder B, if \(h_A = 17\) inches, then \(h_B=2h_A\). So \(h_B = 34\) inches.

Step4: Calculate the height of the combined cylinder

When the two cylinders are glued together along a circular base, the height of the resulting cylinder is \(h=h_A + h_B\). Substitute \(h_A = 17\) and \(h_B = 34\), we get \(h=17 + 34=51\) inches.

Answer:

D. 51