A farmer F_(1) has a land in the shape of a triangle with vertices at P(0,0),Q(1,1) and R(2,0). From this land, a neighbouring farmer F_(2) takes away the region which lies between the side PQ and a curve of the form y=x^(n)(n 1). If the area of the region taken away by the farmer F_(2) is exactly 30% of the area of /_\PQR, then the value of n is

Question

A farmer  F_(1) has a land in the shape of a triangle with vertices at  P(0,0),Q(1,1) and  R(2,0). From this land, a neighbouring farmer  F_(2) takes away the region which lies between the side  PQ and a curve of the form  y=x^(n)(n > 1). If the area of the region taken away by the farmer  F_(2) is exactly  30% of the area of  /_\PQR, then the value of  n is

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Calculate Triangle Area: Calculate the area of triangle PQR. The area of a triangle with vertices at (x1, y1), (x2, y2), and (x3, y3) is given by the formula: Area = 1/2 |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)| Substitute the given points P(0,0), Q(1,1), and R(2,0) into the formula. Area = 1/2 |0(1 - 0) + 1(0 - 0) + 2(0 - 1)| Area = 1/2 |0 + 0 - 2| Area = 1/2 |-2| Area = 1Calculate 30%: Calculate 30% of the area of triangle PQR. 30% of the area is 0.3 times the area of the triangle. 0.3 * Area = 0.3 * 1 = 0.3Set up Integral: Set up the integral to find the area between PQ and y=x^n. The area taken away by farmer F_(2) is the area under the curve y=x^n from x=0 to x=1 minus the area of the triangle formed by the line y=x and the x-axis from x=0 to x=1. The area under the curve is given by the integral from 0 to 1 of x^n dx. The area of the small triangle is 1/2 * base * height = 1/2 * 1 * 1 = 1/2. So, the area taken away is the integral minus the area of the small triangle.Calculate Integral: Calculate the integral and set it equal to 30% of the area of triangle PQR. The integral of x^n from 0 to 1 is (1/(n+1))x^(n+1) evaluated from 0 to 1. This equals (1/(n+1)) * 1^(n+1) - (1/(n+1)) * 0^(n+1) = 1/(n+1). The area taken away is 1/(n+1) - 1/2. Set this equal to 0.3 (from Step 2) and solve for n. 1/(n+1) - 1/2 = 0.3Solve for n: Solve the equation for n. 1/(n+1) - 1/2 = 0.3 Multiply through by 2(n+1) to clear the denominators: 2 - (n+1) = 0.6(n+1) 2 - n - 1 = 0.6n + 0.6 1 - n = 0.6n + 0.6 1 - 0.6 = 0.6n + n 0.4 = 1.6n n = 0.4 / 1.6 n = 1/4

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