Problem 9 - Basic Engineering Mathematics

Question

Determine the diameter and circumference of a circle if an arc of length \(4.75 \mathrm{~cm}\) subtends an angle of \(0.91\) radians.

Step-by-Step Solution

Answer

Diameter: 10.44 cm; Circumference: 32.78 cm.

Step 1: Understand the Arc Length Formula

The formula for arc length is given by the equation \[ s = r \theta \]where \( s \) is the arc length, \( r \) is the radius, and \( \theta \) is the angle in radians. We are given \( s = 4.75 \) cm and \( \theta = 0.91 \) radians.

Step 2: Solve for the Radius

To find the radius, rearrange the arc length formula to solve for \( r \):\[ r = \frac{s}{\theta} \]Substitute the known values into the equation:\[ r = \frac{4.75}{0.91} \approx 5.22 \text{ cm} \]

Step 3: Calculate the Diameter

The diameter \( d \) of the circle is twice the radius. So,\[ d = 2r = 2 \times 5.22 \approx 10.44 \text{ cm} \]

Step 4: Calculate the Circumference

The circumference \( C \) can be found using the formula:\[ C = 2 \pi r \]Substitute the radius:\[ C = 2 \pi \times 5.22 \approx 32.78 \text{ cm} \]

Key Concepts

Arc Length

The arc length of a circle is a portion of the circle’s circumference that corresponds to a given angle. Imagine a slice of pizza – the crust on the edge is like the arc. The length of this arc is mathematically determined by the formula \( s = r \theta \), where \( s \) is the arc length, \( r \) is the radius of the circle, and \( \theta \) is the angle in radians.
In simpler words, the formula combines the radius of the circle and the angle to determine how long the arc is.

• For example, if you have a small pizza slice (a small angle), the arc will be shorter.
• If the radius is larger, like a bigger pizza, the arc will be longer.

In the problem example, the arc length is 4.75 cm corresponding to an angle of 0.91 radians. By rearranging the formula to solve for \( r \), we find that the radius is about 5.22 cm.

Radius and Diameter

The radius is a crucial component in circle geometry. It is the distance from the center of the circle to any point on its edge. Think of the radius as the spoke of a wheel, stretching from the hub to the rim.
When you know the radius, finding the diameter is straightforward: it's simply double the radius. This relationship is expressed with the formula: \( d = 2r \).

• In this exercise, we computed the radius to be approximately 5.22 cm.
• Thus, doubling this gives us a diameter of approximately 10.44 cm.

Understanding these concepts helps you visualize and calculate other properties of the circle more accurately. With complete knowledge of the radius and diameter, you're well-equipped to tackle further problems in circle geometry.

Circumference of a Circle

The circumference is the total distance around the edge of the circle – similar to how you'd measure the perimeter of a square. For any circle, this distance is given by the formula \( C = 2 \pi r \) or equivalently \( C = \pi d \).
The constants in these formulas, \( \pi \), represents the unique ratio of the circumference to the diameter and is approximately 3.14159.

• Using our previously found radius of 5.22 cm, the calculation becomes \( C = 2 \pi \times 5.22 \approx 32.78 \) cm.
• This length, 32.78 cm, represents traveling all the way around the circle once.

When you know either the radius or diameter, determining the circle’s entire surrounding distance becomes a quick process using these simple formulas.