Find two positive numbers that satisfy the given requirements. The second number

Question

Find two positive numbers that satisfy the given requirements. The second number is the reciprocal of the first number and the sum is a minimum. A man is in a boat 2 miles from the nearest point of the coast. He is to go to point Q, located 3 miles down the coast and 1 mile inland (see figure below). He can row at a rate of 2 miles per hour and walk at 4 miles per hour. Toward what point on the coast should he row in order to reach point Q in the least time? A farmer plans to enclose a rectangular pasture adjacent to a river, (see figure). The pasture must contain 45,000 square meters in order to provide enough grass for the herd. What dimensions will require the least amount of fencing if no fencing is needed along the river? Find the point on the graph of the function that is closest to the given point. f(x) = x2 (54,1/2) A rectangle is bounded by the x-axis and the semicircle y = 9-x2 (see figure). What length and width should the rectangle have so that its area is a maximum?

Explanation / Answer

1) y = 1/x

and we have to minimize the sum = x + 1/x

by differentiating, we get

1 - 1/x^2, which when equated to zero, gives , x = 1,-1

and the sum is minimum at x = -1, that is -2

first number = -1

second number = -1

2) We have the total time

= sqrt(4+x2)/2 + sqrt(1+(3-x)2)/4

On differentiating, we get

(2*x)/(4*(sqrt(4+x2)) = (2*(3-x))/(8*sqrt(1+(3-x)2)

On solving for x, we get x = 1 mile

3) we have length to be fenced = x+2y

and the given area = xy = 45000

fencing length has to be minimized, so differentiating the same, we get

f(x) = x + 2*45000/x

= x + 90000/x

f'(x) = 1 - 90000/x^2,

or x = 300 m

y = 150 m

4) the distance between the given point (54,1/2) and the point on curve (x,x^2)

is sqrt((x-54)2 + (x2-1/2)2)

On differentiating we get

4x(x2 - 1/2) + 2(x - 54) = 0

or 4x3 = 108

x3 = 27

x = 3

hence the point is

x = 3, y = 9

5) If we take 2x as the length and y = sqrt(9-x2)

then area is 2x*sqrt(9-x2)

on differentiating, we get

sqrt(9-x2) - 2x2*sqrt(9-x2)

or 1 = 2x2

x = 1/sqrt(2) (smaller value)

y = sqrt(17/2) (larger value)

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