Hi, everyone. Welcome back to another one of our blogs for Harrow School. Today, we’re tackling another 13+ entrance exam question found in Harrow School’s maths paper. This question involves understanding the area a goat can graze when tied to a corner of a square shed. So, let’s dive into it.
Problem Statement
A farmer ties his goat to the outside corner of the shed in his field. The shed is square-shaped in plan view, with each side measuring three metres. If the length of the rope attaching the goat to the shed is one metre, then find the area of the ground that the goat can graze.
Step-by-step Solution
To solve this problem, we’ll follow these steps:
- Visualising the setup
- Understanding the area the goat can graze
- Calculating the area
Let’s break down the steps.
1. Visualising the Setup
First, let’s draw a picture of the shed. The shed is a square with each side measuring three metres.
Next, we tie the rope, which is one metre long, to one corner of the shed. Now, envision the goat tied to the end of this rope.
2. Understanding the Area the Goat Can Graze
With a one-metre rope, the goat can graze in a circular area around that corner. However, the shed itself blocks a quarter of this circle because it occupies that space.
The area the goat can actually graze is three-quarters of a full circle with a radius of one metre.
3. Calculating the Area
The formula to calculate the area of a circle is:
$$A = \pi r^2$$
Given that the radius (r) is one metre, the area of the full circle the goat could graze would be:
$$A = \pi \times 1^2 = \pi \ \text{square metres}$$
Since the shed occupies one-quarter of this area, the actual grazable area is three-quarters of the full circle area:
$$\text{Grazable Area} = \frac{3}{4} \times \pi = \frac{3\pi}{4}$$
So, the area the goat can graze is:
Three-quarters π (π) square metres.
We can write this as:
$$A = \frac{3\pi}{4} \ \text{square metres}$$
Alternate Rope Length: 4 Metres
Now, let’s consider a scenario where the rope is four metres long. The process is similar but with a different radius.
Visualising the Setup
Draw the same square shed, but this time with a four-metre rope.
Understanding the Grazable Area
With a four-metre radius, the goat can graze in a much larger circle, but again, one-fourth of this circle is blocked by the shed.
Calculating the Area for Four Metres
Using the formula for the area of a circle:
$$A = \pi r^2$$
For a four-metre radius, the full circle area is:
$$A = \pi \times 4^2 = 16\pi \ \text{square metres}$$
And three-quarters of this new area:
$$\text{Grazable Area} = \frac{3}{4} \times 16\pi = 12\pi$$
So, the area the goat can graze in this case is:
12π square metres
We can write this as:
$$A = 12\pi \ \text{square metres}$$
Final Summary
Through these calculations, we have determined that:
- With a one-metre rope, the goat can graze an area of (\frac{3\pi}{4}) square metres.
- With a four-metre rope, the goat can graze an area of (12\pi) square metres.
This problem is a classic example demonstrating the practical application of circle area calculations in real-world scenarios. Understanding these basic principles is essential for tackling more complex geometric problems in the future.
Be sure to check out our previous questions, and stay tuned for more from Harrow School entrance exam series.