1. A rectangular box is to be constructed from 2 different materials. The box will have a square base and open top. The material for the bottom costs $4.25/m2. The material for the sides costs $2.5/m2. Find the dimensions of the box with the largest volume if the budget is $500 for the material.
2. The concentration, C, of a drug injected into the bloodstream t hours after injection can be modeled by C(t) = (t/4) + 2t^-2. Determine when the concentration of the drug is increasing and when it is decreasing.
Let the base of the box by x by x m, and let the height be y m.
visualize the box flattened out, the base is a square, with an x by y rectangle attached to each of its sides.
Your "cost" equation is
500 = 4.25x^ + 4*2.5xy
solve this for y
Your main equation, the part that is to be maximized, is
Volume = x^2*y
plug in the y from the other equation, simplify, then differentiate
set the derivative equal to zero and solve.
This is a standard and easy optimization problem and you should not have any difficulty with it.
for the second one, "something" is increasing when its first derivative is positive, and "decreasing" when its first derivative is negative.
you should be able to take it from there.
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