Video Transcript
A goat is tethered by a 10-meter-long rope to the corner of a barn. What area of the field can the goat reach?
Let’s begin by considering what information the diagram gives us. We have a barn with dimensions five meters by 12 meters. We are told that the goat is tethered to the corner of this barn. Let’s place a point on that corner and label it C. The length of the rope connecting the goat to the corner is 10 meters long. If the barn was not there, this would be a much easier problem. We know that the farthest that the goat can be from the corner of the barn is 10 meters. The goat can pull the rope completely straight in any direction, and this represents the radius of a circle. In fact, this is how we get the definition of a circle, which is the set of all points a set distance from the center point.
On the circle we’ve sketched, we can place a few examples of where the goat might be located anywhere around the circumference of a circle with a radius of 10 meters. Now, of course, the goat is not always pulling the 10-meter rope completely straight, so he is free to wander anywhere within the circle. Let’s recall the formula for the area of a circle, 𝐴 equals 𝜋 times radius squared. So if we needed to find the area of a circle with a radius of 10 meters, we’ll substitute 10 for the variable 𝑟, and that gives us 100𝜋 meters squared. Unfortunately, this cannot be our final answer because the barn is overlapping the area of this circle.
Now, we are going to assume that the sides of the barn create a 90-degree angle. The short side of the barn is only five meters, which is half of the length of the radius. Now, we can’t just subtract the rectangular area of the barn because as we can see part of the barn is outside of our circle. If we knew the area of the region that we have shaded in pink, we could subtract that from the area of the circle, which was 100𝜋 meters squared. However, this would not be an easy task because one side of the pink-shaded region is curved. We know the formula for the area of a rectangle is length times width, but this is no longer a rectangle. So we will need to move on to some other more creative ideas.
Let’s clear some space so that we can sketch a new diagram. But let’s also not forget that the area of the circle with a radius of 10 meters is 100𝜋 meters squared. This seems like an important piece of information that we might be able to use later. Let’s sketch our circle over the image we were given. We want a radius of 10 meters. We know that the short end of the barn is five meters, so if we double that length, then we will get the correct radius. We can sketch a smaller model of this circle off to the side to help us think through a few different possible ideas. Let’s consider what would happen if the short edge of the barn was 10 meters instead of five meters. If this were the case, the area that we have shaded in orange would represent all of the area that the goat could reach.
We should recall that a circle contains 360 degrees, and if we divide that into four parts, we get four 90-degree angles. One of those 90-degree angles was the corner of our barn. We can find the area of three-fourths of this circle by multiplying 100𝜋 times three-fourths. This gives us 300𝜋 over four, or simplified that’s 75𝜋 meters squared. Now, let’s return back to the image that we were given. If we follow along the short side of the barn, we see that the other corner of the barn is halfway across the radius. Let’s put a point there named B.
If the goat walked from point C to point B and then turned around the corner, he would only have five meters of rope left. So there’s something that we did not think about earlier. And that is the fact that the goat cannot reach all of the grass behind the barn. If he wants to walk around the corner from point B, he only has five meters in any direction. This means that we could sketch a smaller circle with radius five. If we calculate the blue area of the smaller circle, then we can add it to the orange area from the larger circle. The blue region makes a 90-degree angle at point B.
We can show that the blue region is one-quarter of the area of the smaller circle. This is because our 90-degree angle divided by 360 degrees equals one-fourth. Using the area formula, we can see that the area of the entire smaller circle is 25𝜋 meters squared. The blue area is one-fourth of 25𝜋. 25𝜋 times one-fourth is 25𝜋 over four meters squared. To answer the question of what area of the field the goat can reach, we’re going to add the orange area plus the blue area. To have a common denominator instead of using 75𝜋, we’ll go back to 300𝜋 over four. So in conclusion, the goat can reach an area of 300𝜋 over four plus 25𝜋 over four, which together is 325𝜋 over four meters squared.
To find this answer, we took three-fourths of the area of a circle with radius 10 and added that to one-fourth of the area of a circle with radius five. Since we were given lengths in meters, the area is in meters squared.
