An oval track is made by erecting semi circles on each end of a 58 m by 116 m rectangle. Find the length of the track and the area enclosed by the track.
length is two sides of the rectangle, plus the circumference of the circle.
area is the rectangle plus a whole circle.
To find the length of the track, we need to determine the circumference of the two semi circles at each end of the rectangle and add it to the length of the rectangle.
The length of a semicircle is half the circumference of a full circle. Let's calculate the circumference of the semicircles first.
To find the circumference of a circle, we use the formula:
C = 2πr, where C is the circumference and r is the radius.
The radius of each semicircle is half the length of the rectangle, which is 116 m / 2 = 58 m.
So, the circumference of each semicircle is:
C1 = 2π(58 m) = 116π m.
Since there are two semicircles, the total circumference of the two semicircles is:
C_total = 2(116π m) = 232π m.
Now, we need to add the length of the rectangle to the total circumference to find the length of the track.
Length of the track = Length of the rectangle + Total circumference of the semicircles
= 116 m + 232π m.
To find the area enclosed by the track, we need to subtract the area of the rectangle from the area enclosed by the track.
The area of a rectangle is given by the formula:
A_rectangle = length × width.
The area of the rectangle is therefore:
A_rectangle = (116 m) × (58 m) = 6728 m².
To find the area enclosed by the track, we subtract the area of the rectangle from the area of the track:
Area enclosed by the track = (Area of the track) - (Area of the rectangle)
= [(Length of the track) × (Width of the track)] - (Area of the rectangle)
= [(116 m + 232π m) × (58 m)] - (6728 m²).
Therefore, the length of the track is 116 m + 232π m, and the area enclosed by the track is (116 m + 232π m) × 58 m - 6728 m².
To find the length of the track, we need to calculate the sum of the lengths of the straight sides and the curved parts.
First, let's find the length of each curved part, which is formed by the semi circle.
1. The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius of the circle.
2. In this case, the diameter of the semi circle is equal to the width of the rectangle, which is 58 m.
3. So, the radius of each semi circle is half of the diameter, which is 58/2 = 29 m.
4. Hence, the length of each curved part is C = 2πr = 2π(29 m) = 58π m.
Now, let's calculate the lengths of the straight parts, which are the sides of the rectangle:
1. The length of the rectangle is 116 m, and the width of the rectangle is the same as the diameter of the semi circle, which is 58 m.
2. Therefore, the length of each straight part is equal to the length of the rectangle, which is 116 m.
Now, we can find the length of the entire track by adding up the lengths of all the parts:
Length of track = 2 × (Length of straight parts) + 2 × (Length of curved parts).
Length of track = 2(116 m) + 2(58π m) = 232 m + 116π m.
To find the area enclosed by the track, we’ll combine the rectangle and the two semi circles:
1. The area of the rectangle is length × width = 116 m × 58 m = 6728 m².
2. The area of each semi circle is given by A = (π × r²)/2, where A is the area and r is the radius.
3. So, the area enclosed by both semi circles is A = (π × (29 m)²)/2 = (841π m²)/2 = 420.5π m².
4. Therefore, the total area enclosed by the track is the sum of the area of the rectangle and the two semi circles:
Total area = Area of rectangle + 2 × Area of semi circle.
Total area = 6728 m² + 2 × 420.5π m².
Please note that the final values for the length of the track and the area enclosed will depend on the value of π used in the calculation.