the inside perimeter of a running track shown in figure is 400m the lenghth of each of the straight portion is 90, and the ends are semicircles, if the track is 14 m wide everywherwe,find the area of the track. also find the length of thee outerboundary of the track
inside length:
400 = 90*2 + 2*pi*r
r = 110/pi
outside length: 90*2 + 2*pi*(r+14) = 400+28pi = 487.96
area of track is 2*90*14 + pi(R^2-r^2)
= 2520 + pi((110/pi+14)^2 - (110/pi)^2)
= 5600+196pi
= 6215.75 m^2
First area ofrunning small rectangle =l×b secon step area oftwo small semi-circle =one complete circle-area of two small circle . Then large circle-area of small circle. Thenarea of large circle+rectangle
To find the area of the track, we need to find the area of the rectangular parts and the areas of the semicircular ends separately, and then add them together.
1. Area of the rectangular parts:
The length of each straight portion is 90 meters, and the width is 14 meters. So, the area of each straight portion is 90 * 14 = 1260 square meters. Since there are two straight portions in the track, the total area of the rectangular parts is 2 * 1260 = 2520 square meters.
2. Area of the semicircular ends:
The track is 14 meters wide everywhere, which means it has a radius of (14/2) = 7 meters. The formula to find the area of a circle is A = πr^2. But we need to find the area of the semicircle, so we divide the result by 2.
The area of one semicircle = (π * 7^2) / 2 = (22/7) * 49 / 2 = 77 square meters. Since there are two semicircular ends in the track, the total area of the semicircular ends is 2 * 77 = 154 square meters.
3. Total area of the track:
The total area of the track is the sum of the areas of the rectangular parts and the semicircular ends.
Total area = Area of rectangular parts + Area of semicircular ends = 2520 + 154 = 2674 square meters.
So, the area of the track is 2674 square meters.
To find the length of the outer boundary of the track, we need to sum the length of the straight portions and the circumference of the semicircles.
1. Length of the straight portions:
Each straight portion has a length of 90 meters, and there are two straight portions in the track. So, the length of the straight portions is 2 * 90 = 180 meters.
2. Circumference of the semicircles:
The circumference of a circle is given by the formula C = 2πr. But since we have semicircles, we divide the result by 2.
The circumference of one semicircle = (2 * π * 7) / 2 = 14π meters. Since there are two semicircular ends in the track, the total length of the circumference of the semicircles is 2 * 14π = 28π meters.
3. Total length of the outer boundary of the track:
The total length of the outer boundary of the track is the sum of the length of the straight portions and the circumference of the semicircles.
Total length = Length of straight portions + Circumference of semicircles = 180 + 28π meters.
So, the length of the outer boundary of the track is 180 + 28π meters.
To find the area of the track, we need to calculate the total area of the rectangular parts and the semicircular ends.
First, let's calculate the area of the rectangular parts:
The inside perimeter of the track is given as 400m. Since the track is 14m wide everywhere, the length of each of the straight portions can be calculated by subtracting the width from the inside perimeter: 90m - 2 * 14m = 62m.
The formula for calculating the area of a rectangle is: Area = length * width.
So, the area of each of the straight portions is: Area_rectangular = 62m * 14m.
Next, let's calculate the area of the semicircular ends:
The diameter of the semicircles is equal to the width of the track, which is 14m. So, the radius of each semicircle is: Radius = width / 2 = 14m / 2 = 7m.
The formula for calculating the area of a semicircle is: Area_semicircle = (π * radius²) / 2.
So, the area of each semicircular end is: Area_semicircular = (π * 7m²) / 2.
To find the total area of the track, we add the areas of the rectangular parts and the semicircular ends:
Total Area = 2 * Area_rectangular + 2 * Area_semicircular.
Now, let's calculate the length of the outer boundary of the track:
The outer boundary is the same as the inside boundary plus twice the width of the track added to the length of the two semicircles:
Outer Boundary = Inside Perimeter + 2 * Width + 2 * (π * radius).
Using the given information, we can substitute the values into the above equations to find the answers.