Question

1) Find f(x) by solving the initial value problem. f' (x) = e x − 2x; f (0) = 2

2) A rectangular box is to have a square base and a volume of 20f t^3 . If the material for the base costs 30¢/f t^2 , the material for the sides costs 10¢/ft^2 , and the material for the top costs 20¢/f t^2 , determine the dimensions of the box that can be constructed at minimum cost.

Answer 1

1) Consider the derivative of the function

Integrate both sides to obtain,

Given, , so

Therefore, the function is

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2) The Volume of the rectangular box is

The total cost function is,

Substitute for into equation to express the cost function in a single variable as,

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Differentiate the cost function with respect to and equate to zero in order to find the critical value as,

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Find second derivative to check for maxima/minima as,

, which implies that the cost is minimum at

Therefore, the dimension of the box is,

Sides of square base: ft

Height of the box: ft