AN ATHLETIC FIELD HAS THE SHAPE OF A 40 YARD BY 100 YARD RECTANGLE WITH A SEMICIRCLE AT EACH END. A RUNNING TRACK THAT IS 10 YARDS WIDE SURROUNDS THE FIELD. FIND THE AREA OF THE ATHLETIC FIELD WITH THE TRACK INCLUDED.
Well, well, well, looks like we have quite the sports enthusiast here! Let's get those math muscles flexing, shall we?
First things first, we need to find the area of the rectangular athletic field. That's a simple one, just multiply the length by the width. So 40 yards multiplied by 100 yards gives us 4000 square yards.
Now, here comes the fun part - the semicircles. Since each end has a semicircle, we'll calculate the area of one and double it. The formula for the area of a circle is πr², but since we only have a semicircle, we'll divide it by 2. The radius of the semicircle is half the width of the field, which is 100 yards. So, the area of each semicircle is (π * 50²) / 2.
Now, to find the total area of both semicircles, we just multiply that by 2. Let's do the math: (π * 50²) / 2 * 2.
Lastly, we need to add the area of the rectangular field to the area of both semicircles. So, our final answer is 4000 square yards + (π * 50²).
And there you have it! The area of the athletic field, including the track, is 4000 square yards + (π * 50²). Just plug in the value of π and do the calculations, and you'll have your answer. Happy running!
To find the area of the athletic field with the track included, we need to subtract the area of the two semicircles at each end of the rectangle.
First, let's find the area of the rectangle:
Area of rectangle = Length * Width
Area of rectangle = 100 yards * 40 yards
Area of rectangle = 4000 square yards
Next, we need to find the area of the two semicircles:
Area of a semicircle = (π * r^2) / 2, where r is the radius
The radius of the semicircle is half of the width of the rectangle, which is 40 yards / 2 = 20 yards.
Area of one semicircle = (π * 20^2) / 2
Area of one semicircle = (π * 400) / 2
Area of one semicircle = 200π square yards
Since there are two semicircles, the total area of the two semicircles is 2 * (200π) = 400π square yards.
Now, let's find the area of the track:
The track surrounds the rectangle and has a width of 10 yards. Therefore, it increases the length and width of the rectangle by 10 yards on each side.
New length of the rectangle = length + 2 * width of the track
New length of the rectangle = 100 yards + 2 * 10 yards
New length of the rectangle = 100 yards + 20 yards
New length of the rectangle = 120 yards
New width of the rectangle = width + 2 * width of the track
New width of the rectangle = 40 yards + 2 * 10 yards
New width of the rectangle = 40 yards + 20 yards
New width of the rectangle = 60 yards
Now, let's find the area of the new rectangle including the track:
Area of new rectangle = new Length * new Width
Area of new rectangle = 120 yards * 60 yards
Area of new rectangle = 7200 square yards
Finally, subtract the area of the two semicircles from the area of the new rectangle to find the area of the athletic field with the track included:
Area of the athletic field with the track = Area of new rectangle - Area of the two semicircles
Area of the athletic field with the track = 7200 square yards - 400π square yards
So, the area of the athletic field with the track included is 7200 square yards - 400π square yards.
To find the area of the athletic field, we need to calculate the combined area of the rectangle and the two semicircles, as well as subtract the overlapping areas of the semicircles.
1. Calculate the area of the rectangle:
The rectangle has dimensions of 40 yards by 100 yards, so the area is: 40 yards * 100 yards = 4000 square yards.
2. Calculate the area of the two semicircles:
Each semicircle has a radius of 40 yards (half of the width) because the rectangle is 40 yards wide.
The formula to calculate the area of a semicircle is: (π * radius^2) / 2.
Therefore, the area of one semicircle is: (3.14 * 40^2) / 2 = 2512 square yards.
The combined area of the two semicircles is: 2 * 2512 = 5024 square yards.
3. Calculate the overlapping areas of the semicircles:
Since the semicircles overlap each other, we need to subtract the area of the overlapping portion.
The overlapping portion is a rectangle, and its dimensions are the width of the track (10 yards) and the diameter of the semicircle (80 yards = 2 * 40 yards radius).
Therefore, the overlapping area is: 10 yards * 80 yards = 800 square yards.
4. Calculate the area of the running track:
The running track surrounds the field, so its area is the area difference between the combined area of the rectangle and the semicircles and the overlapping area of the semicircles: (4000 + 5024) - 800 = 8224 square yards.
Hence, the area of the athletic field with the track included is 8224 square yards.
you have a rectangle that is 100x(40+20)
and a circle that has diameter 40+20=60
so, just add those two areas.
If the area inside the circular ends but outside the rectangle does not count, then you have to subtract off the smaller circle of diameter 40.