Question

guided practice
what is the surface area of the cylinder?
give your answer in terms of \\(\pi\\).
\\(\text{cm}^2\\)
(image of a cylinder with diameter 3 cm and height 4 cm)

Explanation:

Step1: Identify cylinder parameters

Radius $r = \frac{3}{2} = 1.5$ cm, height $h = 4$ cm

Step2: Calculate surface area

Surface area = $2\pi r(r + h) = 2\pi×1.5×(1.5 + 4) = 3\pi×5.5 = 16.5\pi$? Wait, no—wait, the diameter is 3 cm, so radius is 1.5 cm. Wait, lateral surface area is $2\pi rh = 2\pi×1.5×4 = 12\pi$. Two circular bases: $2×\pi r² = 2×\pi×(1.5)² = 2×\pi×2.25 = 4.5\pi$. Total surface area: $12\pi + 4.5\pi = 16.5\pi$? Wait, maybe I misread—wait, the problem says "surface area of the cylinder"—if it's total surface area (including two bases), yes: $2\pi r(r + h) = 2\pi×1.5×(1.5 + 4) = 3\pi×5.5 = 16.5\pi$. But wait, maybe the diameter is 3 cm so radius 1.5, height 4. Correct total surface area is $2\pi r² + 2\pi rh = 2\pi(1.5)² + 2\pi(1.5)(4) = 4.5\pi + 12\pi = 16.5\pi$. But wait, maybe the problem expects using diameter directly? No, radius is half diameter. So final answer is $16.5\pi$ or $\frac{33}{2}\pi$. But let's check again: $r = 3/2 = 1.5$, $h=4$. Total surface area: $2\pi(1.5)^2 + 2\pi(1.5)(4) = 2\pi(2.25) + 12\pi = 4.5\pi +12\pi=16.5\pi$. Yes.

Wait, maybe I made a mistake earlier—so the correct answer is $16.5\pi$ cm² or $\frac{33}{2}\pi$ cm².

Step1: Find radius

Radius $r = \frac{3}{2} = 1.5$ cm

Step2: Calculate base area sum

$2×\pi r² = 2×\pi×(1.5)² = 4.5\pi$ cm²

Step3: Calculate lateral surface area

$2\pi rh = 2\pi×1.5×4 = 12\pi$ cm²

Step4: Total surface area

$4.5\pi +12\pi =16.5\pi$ cm²

Answer:

$21\pi$ cm²