Consider the amount produced, az, as a function of the labor used, I, and the ma

Question

Consider the amount produced, az, as a function of the labor used, I, and the materials used, m. Let us suppose that the amount is the number of units produced, on average, in one day, that labor is measured in the number of hours used in a day, and that material is measured in the number of pounds used in a day. Assume that 10.75 m0.5 Suppose that labor costs 15 dollars an hour and the material costs 5 dollars a pound. 1. Suppose that 1000 dollars are used to buy labor and materials. Using a substitution, find the values of l and m that maximize production, and find the maximum value of z. That is, solve for either variable in 151 + 5m 1000, find z as a function of the remaining variable, and maximize using one-variable calculus

Explanation / Answer

Given

15l + 5m = 1000

=> 3l+m = 200 (Divide by 5)

=> m = (200 - 3l )

x = l0.75m0.5

=>x4 = l3(200 - 3l)2

For x to be maximum dx/dl = 0

4x3dx/dt = 0 => 3l2(200-3l)2 + l3 *2*(200-3l) * -3 = 0

=> (200-3l) = l *2

=> 5l = 200

=> l = 40

m = 200 - 3l = 200 - 3*40 = 80

x max = ( l3m2)1/4 = (403 * (80)2 )1/4 = 142.26

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