A calf is tied with a rope of length 6m at the corner of a square grassy lawn of side 20m. If the length of the rope is increased by 5.5m, find the increase in area of grassy lawn in whi the calf can graze.
Let the calf be tried at the corner A of the square lawn.Then, the increase in area = difference of two sectors of central angle 90 each and radii 11.5m (6m+5.5m) and 6m, which is shaded region in figure so,required increase in area = [90/360into pia into( 11.5)into 11.5-90/360into pia into6into ]msq
=75.625m.sq
No idea
To find the increase in the area of the grassy lawn in which the calf can graze, we need to determine the area of the annular region formed when the length of the rope is increased.
Let's break down the problem step by step:
Step 1: Find the area of the initial grazing region.
The initial grazing region is a circle with a radius equal to the length of the rope. Since the rope length is 6m, the radius of the initial grazing region is also 6m.
The formula to calculate the area of a circle is:
A = π * r^2, where A is the area, and r is the radius.
Therefore, the initial grazing area is:
A_initial = π * (6)^2 = π * 36
Step 2: Find the area of the larger grazing region.
When the length of the rope is increased by 5.5m, the new radius of the grazing region is 6 + 5.5 = 11.5m.
The formula to calculate the area of the larger grazing region is the same as for the initial grazing region:
A_new = π * (11.5)^2 = π * 132.25
Step 3: Calculate the increase in the area of the grazing region.
To find the increase in the grazing area, subtract the initial grazing area from the new grazing area:
Increase in area = A_new - A_initial
Increase in area = (π * 132.25) - (π * 36)
Increase in area = π * (132.25 - 36)
Step 4: Simplify the expression.
Increase in area = π * 96.25
So, the increase in the area of the grassy lawn in which the calf can graze is approximately 96.25π square meters.
Thanx
Did you make a sketch?
The area reached by the horse is 1/4 of a circle with original radius of 6 m
that area = (1/4)π(36) = 9π m^2
after lengthening of rope, r = 11.5 m
new area for grazing = (1/4)π(11.5)^2 m^2
= 33.0625π m^2
So what is the increase in area ?