Problem:
A cow is tied with a rope of length 7 cm at the corner of a triangular field with each side measuring 15 cm. If the length of the rope is increased by 4.5 m, find the increase in the area of the field in which the cow can graze.

Solution:


To solve this problem, we need to follow these steps:

Step 1: Find the area of the original triangular field
The triangular field is an equilateral triangle with each side measuring 15 cm. To find the area of an equilateral triangle, we can use the formula:


Area = (sqrt(3)/4) * (side length)^2


Plugging in the values, we get:


Area = (sqrt(3)/4) * (15)^2


Area = (sqrt(3)/4) * 225


Area = (1.732/4) * 225


Area = 3.464 * 225


Area = 781.5 cm^2

Step 2: Find the area of the larger circular field
When the length of the rope is increased by 4.5 m, the total length becomes 7 + 4.5 = 11.5 m. This length represents the radius of the circular field in which the cow can graze. To find the area of a circle, we can use the formula:


Area = π * (radius)^2


Plugging in the values, we get:


Area = π * (11.5)^2


Area = π * 132.25


Area = 413.25π m^2

Step 3: Find the increase in area of the field
The increase in area of the field in which the cow can graze is the difference between the area of the larger circular field and the area of the original triangular field. We can calculate this by subtracting the area of the triangular field from the area of the circular field:


Increase in area = Area of circular field - Area of triangular field


Increase in area = 413.25π - 781.5 cm^2

Step 4: Calculate the final answer
To calculate the final answer, we need to convert the area from square meters to square centimeters. Since 1 square meter is equal to 10,000 square centimeters, we can multiply the area in square meters by 10,000:


Increase in area = 413.25π - 781.5 cm^2


Increase in area = 413.25π - 781.5 * 10000 cm^2


Increase in area = 413.25π - 7815000 cm^2

Result:
The increase in the area of the field in which the cow can graze, when the length of the rope is increased by 4.5 m, is 413.25π - 7815000 cm^2.