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Chapter 5: Problem 27

Suppose a slice of pizza with an angle of \(\frac{5}{6}\) radians has an area of 21 square inches. What is the diameter of this pizza?

Short Answer

Expert verified
The diameter of the pizza is approximately 11.46 inches.

Step by step solution

01

Write down the formula for the area of a sector

We will use the formula for the area of a sector of a circle, which is given by: \(A = \frac{1}{2}r^2\theta\), where \(A\) is the area of the sector, \(r\) is the radius of the circle, and \(\theta\) is the angle of the sector in radians.
02

Plug in the given values for area and angle into the formula

We are given the area, \(A = 21\) square inches, and the angle, \(\theta = \frac{5}{6}\) radians. Plugging these values into the formula, we get: \(21 = \frac{1}{2}r^2 \cdot \frac{5}{6}\).
03

Solve the equation for the radius, r

To solve for \(r\), we will first multiply both sides of the equation by \(\frac{6}{5}\) to cancel out the \(\frac{5}{6}\) in the equation: \(\frac{6}{5} \cdot 21 = r^2 \cdot \frac{1}{2}\). Next, simplify the left side of the equation: \(\frac{6 \cdot 21}{5} = \frac{1}{2}r^2\). Now, multiply both sides of the equation by 2 to get rid of the \(\frac{1}{2}\) on the right side: \(2 \cdot \frac{6 \cdot 21}{5} = r^2\). This simplifies to: \(\frac{12 \cdot 21}{5} = r^2\). To find the radius, take the square root of both sides: \(r = \sqrt{\frac{12 \cdot 21}{5}}\).
04

Calculate the value of the radius

Now, simply carry out the calculation to find the value of the radius: \(r = \sqrt{\frac{12 \cdot 21}{5}} \approx 5.73\).
05

Find the diameter of the pizza

Since the diameter of a circle is twice the radius, simply multiply the radius by 2: \(d = 2 \cdot 5.73 \approx 11.46\). So, the diameter of this pizza is approximately 11.46 inches.

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