Application of differentiation (a) A farmer wants to fence an area of 1, 500 squ

Question

Application of differentiation (a) A farmer wants to fence an area of 1, 500 square metre in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can she do this so as to minimize the cost of the fence? (b) A cone-shaped drinking cup is made from a circular piece of paper of radius by cutting out a sector and joining the edges and. Find the maximum capacity of such a cup. (c) The formula for the power output P of a battery is P = VI - RI^2, where V is the electromotive force in volts, R is the resistance in Ohms, and I is the current in amperes. Find the current that corresponds to a maximum value of P in a battery, for which V = 12 volts and R = 0.5 Ohm. Assume that a 15-ampere fuse bounds the output in the interval 0 lessthanorequalto I lessthanorequalto 15.

Explanation / Answer

(a)

let the sides be a and b ( a is the smaller side)

given ab = 1500

to minimize (f) = 2(a+b) + a = 2 (a + 1500/a) + a

df/da = 2(1-1500/a2) + 1 = 0 (first derivative is 0 at the local minima)

(1-1500/a2) = -0.5

1.5 = 1500/a2

a = 10

b = 150

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