Problem 176 - Calculus Volume 1

Question

A total of \(250,000 \mathrm{m}^{2}\) of land is needed to build a nuclear power plant. Suppose it is decided that the area on which the power plant is to be built should be circular. a. Find the radius of the circular land area. b. If the land area is to form a 45° sector of a circle instead of a whole circle, find the length of the curved side.

Step-by-Step Solution

Answer

Radius is approximately 282.16 m; arc length is approximately 221.71 m.

Step 1: Understand the Problem

The exercise gives you a total land area of \(250,000 \, \mathrm{m}^2\) that is circular and asks for the radius and the length of the curved side if it is a sector of 45°.

Step 2: Area of a Circle Formula

To find the radius of a circle, use the formula for the area of a circle, which is \(A = \pi r^2\), where \(A\) is the area and \(r\) is the radius.

Step 3: Solve for Radius

Given \(A = 250,000 \, \mathrm{m}^2\), set \( \pi r^2 = 250,000 \. \mathrm{m}^2\). Solve for \(r\):\[r = \sqrt{\frac{250,000}{\pi}}\]

Step 4: Calculate the Radius

Using a calculator, determine \(r\):\[r \approx \sqrt{\frac{250,000}{3.14159}} \approx \sqrt{79,577.47} \approx 282.16 \, \mathrm{m}\]

Step 5: Policy Shift to Sector Area

For a sector forming part of a circle, note that the area proportion changes. The question asks for a sector of 45° with the same central angle length on the curved part.

Step 6: Find Circumference of Whole Circle

The circumference \(C\) of the whole circle is given by \(C = 2\pi r\). Substitute \(r = 282.16 \, \mathrm{m}\) into the formula to find \(C\). Foot your substitutions and calculations accurately.

Step 7: Calculate Circumference

Now calculate \(C\):\[C = 2 \times \pi \times 282.16 \approx 1773.64 \, \mathrm{m}\]

Step 8: Sector's Arc Length

Determine the arc length of the 45° sector. The arc length is the fraction of the circle's circumference corresponding to the angle:\[\text{Arc Length} = \frac{45}{360} \times 2\pi r = \frac{45}{360} \times 1773.64\]

Step 9: Calculate the Arc Length

Simplify the term above:\[\text{Arc Length} \approx \frac{1}{8} \times 1773.64 \approx 221.71 \, \mathrm{m}\].

Key Concepts

Radius Calculation

Finding the radius of a circle is an essential aspect of understanding circles. To find the radius when given the area of the circle, we use the formula for the area of a circle:

• \( A = \pi r^2 \)
• Here, \( A \) stands for the area and \( r \) is the radius.

Given a circular area of \( 250,000 \, \mathrm{m}^2 \), we aim to find the radius.Start by setting up the equation with the known area:

• \( \pi r^2 = 250,000 \).

To solve for \( r \), rearrange the formula:

• \( r = \sqrt{\frac{250,000}{\pi}} \).

With the help of a calculator, you substitute \( \pi \approx 3.14159 \) and find:

• \( r \approx \sqrt{\frac{250,000}{3.14159}} \approx 282.16 \, \mathrm{m} \).

Thus, the radius is approximately \( 282.16 \, \mathrm{m} \). This step is crucial for further calculations like finding the circumference or other properties of the circle.

Sector of a Circle

A sector of a circle is a part of a circle, which resembles a slice of the entire pie. It consists of two radii and the arc between them.
The angle formed between these two radii is called the central angle. In our example, we have a sector of 45°.For calculating properties of a sector, understanding the proportional relationship to the whole circle is key:

• The area of a sector: \( \text{Sector area} = \frac{\text{central angle}}{360°} \times \pi r^2 \).
• The length of the arc (curved side) of the sector: \( \text{Arc length} = \frac{\text{central angle}}{360°} \times C \), where \( C \) is the circumference.

As only a 45° slice is being considered, each calculated property for the sector will be \( \frac{1}{8} \) (since \( \frac{45}{360} = \frac{1}{8} \)) of the whole circle. This understanding helps in narrowing down calculations precisely for sectors with different angles.

Arc Length

The arc length refers to the distance along the curved edge of a sector. It is a portion of the total circumference of the circle. For a 45° sector, this means calculating \( \frac{1}{8} \) of the circle's full circumference since 45° is one eighth of 360°.First, calculate the full circumference of the circle using the formula:

• \( C = 2\pi r \)

Using the radius found previously, \( r = 282.16 \mathrm{m} \), we get:

• \( C \approx 2 \times 3.14159 \times 282.16 \approx 1773.64 \, \mathrm{m} \)

To find the arc length of our sector:

• \( \text{Arc Length} = \frac{45}{360} \times 1773.64 \approx \frac{1}{8} \times 1773.64 \approx 221.71 \, \mathrm{m} \)

This reveals the length of the curved side of the sector. Remember, arc length is always just a fraction of the entire circumference based on the central angle.