Question

Around a circular garden a circular road is to be repair which costs Rs. 22176 at the rate of Rs. 1 per sq. \(\mathrm{m}\). If the inner radius is \(112 \mathrm{~m}\), find the width of the circular road : (a) \(18 \mathrm{~m}\) (b) \(28 \mathrm{~m}\) (c) \(14 \mathrm{~m}\) (d) none of these

Short answer

Answer: (a) 18 m

Step-by-step solution

Find the area of the circular road

The cost of repairing the road is given as Rs. 22176 and the rate per square meter is Rs. 1. We can find the area of the circular road by dividing the cost by the rate: Area = Cost / Rate Area = 22176 / 1 Area = 22176 \(\mathrm{m^2}\)

Find the area of the inner circle

The inner radius of the circular road is given as 112m. We can find the area of the inner circle by using the formula for the area of a circle: Area = \(\pi r^2\), where \(r\) is the radius of the circle Inner circle area = \(\pi (112)^2\) Inner circle area = 39424\(\pi\) \(\mathrm{m^2}\)

Find the area of the outer circle

The area of the circular road is the difference between the areas of the outer circle and the inner circle. Therefore, we can find the area of the outer circle by adding the area of the circular road to the area of the inner circle: Outer circle area = Area of the circular road + Inner circle area Outer circle area = 22176 + 39424\(\pi\) Outer circle area = 39424\(\pi\) + 22176

Find the radius of the outer circle

Now, we can use the formula for the area of the circle to find the radius of the outer circle: Outer circle area = \(\pi R^2\), where \(R\) is the radius of the outer circle 39424\(\pi\) + 22176 = \(\pi R^2\) Divide both sides by \(\pi\): R^2 = 39424 + \(22176/\pi\) Now, take the square root of both sides to find the value of R R = \(\sqrt{39424 + 22176/\pi}\) R ≈ 130

Find the width of the circular road

The width of the circular road is the difference between the outer circle radius and the inner circle radius: Width = R - r Width = 130 - 112 Width = 18 meters So, the width of the circular road is 18 meters. The correct answer is (a) \(18 \mathrm{~m}\).

Key concepts

Circle Area Calculation

Calculating the area of a circle is a fundamental concept in geometry. The formula to find the area is \( \pi r^2 \), where \( r \) represents the radius of the circle. The symbol \( \pi \) (pi) is a constant approximately equal to 3.14159. To calculate the area:

• Identify the radius of the circle.
• Square the radius (multiply it by itself).
• Multiply the squared radius by \( \pi \).

For example, if a circle has a radius of 112 meters, its area would be \( \pi \times 112^2 = 39424\pi \) square meters. This formula helps in determining larger calculations like how much space a circle covers.

Radius and Diameter

Understanding the relationship between radius and diameter is crucial when dealing with circles. The radius is the distance from the center of the circle to any point on its edge. The diameter is twice the radius, representing the longest line that passes through the center, connecting two points on the circle.

• If radius \( r \) is known, then diameter \( d = 2r \).
• If the diameter is given, calculate the radius by dividing the diameter by 2.

This basic knowledge is useful in many problems, such as finding a circle's area or the width of a circular path.

Problem-Solving Steps

Solving geometry problems often involves systematic steps. Let's break down the steps used in solving the width of a circular road problem:

• First, find the area of the circular road by dividing the cost by the rate per square meter.
• Next, calculate the area of the inner circle using its radius.
• Combine these to find the outer circle's area, then use this area to determine the outer radius.
• Calculate the width of the circular road by subtracting the inner radius from the outer radius.

By following these structured steps, you can solve similar problems efficiently, ensuring no important detail is missed.