Problem 69 - Precalculus with Limits

Question

CIRCUMSCRIBED AND INSCRIBED CIRCLES In Exercises 68 and 69, use the results of Exercise 67. Find the length of the largest circular running track that can be built on a triangular piece of property with sides of lengths 200 feet, 250 feet, and 325 feet.

Step-by-Step Solution

Answer

The length of the largest circular running track that can be built on the triangular piece of property is approximately 506.707 feet.

Step 1: Calculate the semi-perimeter of the triangle

The semi-perimeter of the triangle is calculated using the formula \( s = \frac{a + b + c}{2} \), where a, b, c are the lengths of the sides. In this case, \( s = \frac{200 + 250 + 325}{2} = 387.5 \) feet.

Step 2: Calculate the area of the triangle

The area of the triangle is calculated using Heron's formula: \( A = \sqrt{s(s - a)(s - b)(s - c)} \). In this case, \( A = \sqrt{387.5(387.5 - 200)(387.5 - 250)(387.5 - 325)} = 31250 \) square feet.

Step 3: Calculate the radius of the inscribed circle

The radius of the inscribed circle is calculated using the formula \( r = \frac{A}{s} \). In this case, \( r = \frac{31250}{387.5} = 80.645 \) feet.

Step 4: Calculate the length of the track (circumference of the circle)

The length of the track is the circumference of the inscribed circle, which can be calculated using the formula \( c = 2 \pi r \). In this case, \( c = 2 \pi 80.645 \approx 506.707 \) feet.

Key Concepts

Semi-Perimeter

To start understanding Heron's formula and the calculation of triangles, we first need to delve into the concept of semi-perimeter. The semi-perimeter of a triangle is essentially half of the perimeter. This is calculated by adding all the side lengths of the triangle and then dividing the sum by two.

For example, if a triangle has sides of length 200 feet, 250 feet, and 325 feet, the formula to find the semi-perimeter (\( s \)) is straightforward:

• Formula: \( s = \frac{a + b + c}{2} \)
• Calculation: \( s = \frac{200 + 250 + 325}{2} = 387.5 \text{ feet} \)

The semi-perimeter plays a critical role in Heron’s formula for calculating the area of a triangle, which in turn helps solve problems involving inscribed circles.

Inscribed Circle

An inscribed circle of a triangle is one that fits perfectly within the triangle, touching all three sides. The center of this circle is the intersection of the triangle's angle bisectors. The radius of the inscribed circle is crucial because it ties together the area of the triangle and the semi-perimeter in a neat formula.

This formula is \( r = \frac{A}{s} \), where \( A \) is the area of the triangle and \( s \) is the semi-perimeter. For a triangle with an area of 31250 square feet and a semi-perimeter of 387.5 feet, the radius calculates as:

• Calculation: \( r = \frac{31250}{387.5} \approx 80.645 \text{ feet} \)

The inscribed circle is instrumental in many geometric constructions and proofs, and understanding its properties and equation is key to mastering triangle geometry.

Circumference of a Circle

The circumference of a circle is the total distance around it, frequently referred to as the circle's perimeter. When considering an inscribed circle in a triangle, this measurement becomes highly relevant, especially if you're looking to determine the size of an object like a circular running track.

To find the circumference of the inscribed circle, you simply use the formula \( c = 2 \pi r \). Given that we calculated the radius of the inscribed circle as approximately 80.645 feet in the previous section, the circumference follows:

• Calculation: \( c = 2 \pi \times 80.645 \approx 506.707 \text{ feet} \)

This length is the largest possible circular track that could be perfectly aligned within the triangle, showcasing how geometric principles can have practical applications in real-world problems.