Question

calculate the surface area of the following. show all your work. round each answer to the nearest tenth. 7.7 mi 19.8 mi 9 mm 6.5 mm (just the label)

Explanation:

Step1: Recall surface - area formula for cylinder

The total surface area of a cylinder $A = 2\pi r^{2}+2\pi rh$, and the lateral - surface area (label area) of a cylinder $A_{l}=2\pi rh$. For the first cylinder with diameter $d = 7.7$ mi, radius $r=\frac{d}{2}=\frac{7.7}{2}=3.85$ mi and height $h = 15.8$ mi. For the second cylinder (label only), with diameter $d = 6.5$ mm, radius $r=\frac{d}{2}=3.25$ mm and height $h = 9$ mm.

Step2: Calculate total surface area of first cylinder

$A_1=2\pi r^{2}+2\pi rh=2\pi(3.85)^{2}+2\pi(3.85)(15.8)$
$=2\pi\times14.8225 + 2\pi\times60.83$
$=29.645\pi+121.66\pi$
$=(29.645 + 121.66)\pi$
$=151.305\pi$
$\approx151.305\times3.14159\approx475.4$ square miles.

Step3: Calculate label area of second cylinder

$A_2 = 2\pi rh=2\pi(3.25)(9)$
$=58.5\pi$
$\approx58.5\times3.14159\approx183.7$ square millimeters.

Answer:

The total surface area of the first cylinder is approximately $475.4$ square miles and the area of the label of the second cylinder is approximately $183.7$ square millimeters.