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Chapter 3: Problem 66

The function \(V\) described by \(V(r)=4 \pi r^{2}\) gives the surface area of a sphere with radius \(r\). Find the surface area when the radius is 5 cm.

Short Answer

Expert verified
The surface area is 314 square cm.

Step by step solution

01

Identify the given values

The problem gives the formula for the surface area of a sphere, which is \[ V(r) = 4 \pi r^2 \]. It also provides the radius, \( r = 5 \) cm.
02

Substitute the radius into the formula

Replace \( r \) in the formula with the given radius (5 cm): \[ V(5) = 4 \pi (5)^2 \]
03

Compute the square of the radius

Calculate \( 5^2 \): \[ 5^2 = 25 \]
04

Multiply by 4 and \( \pi \)

Substitute \( 25 \) back into the equation: \[ V(5) = 4 \pi \times 25 \]Simplify the multiplication first: \[ V(5) = 100 \pi \]
05

Calculate the surface area

If a numerical value for \( \pi \) is needed, use \( \pi \approx 3.14 \): \[ V(5) = 100 \times 3.14 = 314 \] Thus, the surface area is 314 square cm.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Surface Area Formula
The surface area formula for a sphere helps us find the total area that covers the outside of a sphere. The formula is given by \( V(r) = 4 \pi r^2 \). This formula tells us how to calculate the surface area (\

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