Problem 46 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. In Exercises \(41-44\) refer to Fig. 4.42. In Exercises \(41-43,\) express the other parts in terms of the listed parts, which are assumed known. Find the exact perimeter of an equilateral triangle is inscribed in a circle (each vertex is on the circle) of circumference \(20 \pi\)

Step-by-Step Solution

Answer

The exact perimeter of the triangle is \(30\sqrt{3}\).

Step 1: Determine the Radius of the Circle

The circumference of the circle is given as \(20\pi\). The formula to find the circumference is \(C = 2\pi r\), where \(r\) is the radius. To find \(r\), we set up the equation \(2\pi r = 20\pi\). Dividing both sides by \(2\pi\), we find \(r = 10\).

Step 2: Relate the Radius to the Side Length

In an equilateral triangle inscribed in a circle, the radius is equal to the circumradius of the triangle. The relationship between the side length \(s\) of the equilateral triangle and its circumradius \(R\) is given by \(R = \frac{s}{\sqrt{3}}\). In this case, \(R\), the radius of the circle, is \(10\).

Step 3: Solve for the Side Length

From \(10 = \frac{s}{\sqrt{3}}\), we solve for \(s\) by multiplying both sides by \(\sqrt{3}\), which gives \(s = 10\sqrt{3}\). This is the length of each side of the equilateral triangle.

Step 4: Calculate the Perimeter

The perimeter \(P\) of an equilateral triangle is three times the side length \(s\). Thus, \(P = 3 \times 10\sqrt{3} = 30\sqrt{3}\). This is the exact perimeter of the inscribed equilateral triangle.

Key Concepts

Circumference of a Circle

To understand the circumference of a circle, envision a circle as a perfectly round shape. The circumference is the total distance around the circle's boundary. It is similar to the perimeter you would measure, but for circles, we call it the circumference.

The formula to calculate the circumference () of any circle is given by:

• \( C = 2\pi r \)

Here, \( r \) represents the radius, which is the distance from the center of the circle to any point on its boundary. In our exercise, the circumference was already known to be \( 20\pi \). This is helpful, as knowing the circumference allows us to calculate the radius, which serves as a stepping stone to solving further parts of our problem. Simply divide the known circumference by \( 2\pi \), resulting in \( r = 10 \). By determining this, we open the door to calculating other important aspects of the problem, like the side lengths of the triangle inscribed inside this circle.

Inscribed Figures

An inscribed figure is a shape drawn inside another, where all vertices touch the boundary of the larger shape. In our exercise, we have an equilateral triangle inscribed in a circle.

This means:

• Each vertex of the triangle touches the circle.
• The circle is known as the circumcircle of the triangle.
• The circle's radius is the distance from the center to each vertex of the inscribed triangle.

For an equilateral triangle, all sides are equal, and when inscribed in a circle, the radius of the circle also becomes the circumradius of the triangle. This relationship allows us to connect the circle with the triangle and solve for unknowns like side lengths.

Side Length Calculation

Calculating the side length of our equilateral triangle involves recognizing its relationship with the circle's radius, assuming they share a center. For any equilateral triangle inscribed in a circle, the following formula applies:

• \( R = \frac{s}{\sqrt{3}} \)

Where \( R \) is the circumradius (or the circle's radius in this context) and \( s \) is the side length of the triangle. Given that the radius is \( 10 \), you substitute \( R \) with \( 10 \) to get:

• \( 10 = \frac{s}{\sqrt{3}} \)

To solve for \( s \), multiply both sides by \( \sqrt{3} \), resulting in \( s = 10\sqrt{3} \). This formula efficiently gives us the precise measurement for each side based simply on the radius of the inscribing circle. It simplifies our process by converting a vital geometric property into a concrete numerical side length.

Perimeter Calculation

Calculating the perimeter of an equilateral triangle involves a straightforward, yet critical, step. Knowing that all three sides of the triangle are unequal measure:

• If side \( s = 10\sqrt{3} \),
• The perimeter \( P \) is simply three times the side length \( P = 3 \times s \).

Thus, our calculation becomes \( P = 3 \times 10\sqrt{3} \), which results in \( P = 30\sqrt{3} \). This gives us the exact perimeter of our inscribed equilateral triangle, showing how circle properties can help us determine linear measurements quickly. The process of finding each triangle's side length from the radius simplifies what might seem a complex task with accessible mathematical concepts.