Maxium area questions?

Question

I am not going to lie. I am lazy and we have a take home test in my calc class and I need to solve these problems:

1. A closed rectangular box with a square base is to be contructed using two different types of wood. The top is made of wood costing $3 per square foot and the remainder is made of wood costing $1 per square food. suppose that $48 is available to spend. Find the dimensions of the box of greatest volume that can be constructed.

2. A small orchard yields 25 bushels of fruit per tree when planted with 40 trees. Because of the overcrowding the yield per tree is reduced by 1/2 bushel for each additional tree that is planted. How many trees should be planted in order to maximize the total yiled of the orchard.

3. A poster is to have an area of 125 square inches. The material is to be surrounded by a margin of 3 inches at the top and 2 at the bottom and sides. Find the dimensions of the poster that maximizes the area of the printed material.

Thank you for the help!

Answer 1

1. Let the dimensions be x,x,h
then volume = x^2 h = V
Cost = 3*x^2 + x^2 + 4xh = 48
=> x^2 + xh = 12
=> xh = 12 - x^2
V = x^2 h = x.x.h = x.(12-x^2) = 12x - x^3
V' = 0 => 12 - 3x^2 = 0 => x = 2 , h = 4
Answer dimensions are 2,2,4

2) Let the additional no of trees be n
then total fruits = 25*40 + 1.(39.5) + 2.(39) + .. n.(40-n/2)
= (25-n/2)(40+n)
= 1000 + 5n -n^2/2
Maximize this function and find the nearest integer value of n which is n = 5

Total no of trees = 45

3)
let the length be x and width y
total area = (x+5)(y+4) = 125
area of printed material = xy = A
xy + 5y + 4x = 105
A = 105 - 5y - 4x = 105 - 5A/x -4x
A(1+5/x) = 105 - 4x
A = x(105-x)/(x+5)
Maximize this function to find x,y, and A

Answer 2

#2 is just high school alg 1:
total yield = yield per tree times number of trees. x = number of additional trees, so

y = (25 - x/2)(40+x)

This parabola opens downward, has 2 x intercepts. When x = 50, yield per tree is 0. If x = -40, number of trees = 0. Peak of curve is halfway between, at x = 5.

Answer 3

set up a formulation.Then sparkling up w/derivatives. So the area of one equivalent rectangle is xy. enable x and y be the lengths and widths. so the completed section is 2xy. oh guy the algebra were given me. nicely, i comprehend that the max section is discovered with a sq., x=y, yet i'm able to't practice it to you proper right here. with any luck yet another answer will. section =2xy=2x^2. Perimeter=2(x+x+x+x)=8x=1200. x=100 and fifty. section=45000 b/c we discovered the fringe for the utmost a threat (a sq.) and plugged lower back into section. it really is nicely the algebra thanks to do it, the calculus way is rather a similar. except you doesn't assume the sq. is the biggest a threat section (which that's). the placement with this question is it asks to outline the area using perimeter, because it doesn't let us know the rectangles have a similar perimeter only similar section. and that i forgot a thanks to do this problem. the guy below me assumed that the scale and widths of both aspects are equivalent. this can introduce extranious ideas.