PART A Find the dimensions of the open rectangular box of maximum volume that ca

Question

PART A

Find the dimensions of the open rectangular box of maximum volume that can be made from a sheet of cardboardby cutting congruent squares from the corners and folding up the sides. Then find the volume.

PART B

Drum Tight Containers is designing an open-top, square-based, rectangular box that will have a volume of 4 in^3. What dimensions will minimize surface area? What is the minimum surface area?

PART C

An apple farm yields an average of 44 bushels of apples per tree when 24 trees are planted on an acre of ground. Each time 1more tree is planted per acre, the yield decreases by 1 bushel (bu) per tree as a result of crowding. How many trees should be planted on an acre in order to get the highest yield?

PART D

A mailing service places a limit of 42 in. on the combined length and girth of(distance around) a package to be sent parcel post. What dimensions of a rectangular box with square cross-section will contain the largest volume that can be mailed? (Hint: There are two different girths.) The dimensions are x=___ and y=____

Explanation / Answer

Solution:(B)

Let's call the lentgh of the base x.

Since the volume = x^2 * h (base area * height), and it must be 4 in^3,

that meas x^2 * h = 4 >>> h = 4/x^2

Now, the surface area is: A = x^2 + 4xh (1 side with the area of the base, since the box is open, 4 sides that are determined by the length of the base and the height).

Plugging in h = 4/x^2, that means A(x) = x^2 + 16/x

We need the minimum surface area. To find it, we can find the derivative of A(x) and equate it to zero:

A'(x) = 2x - 16/x^2 = 0
==> 2x = 16/x^2
==> x^3 = 8 >>> x = 2,

=> h = 4/x^2 = 4/4 = 1

Hence, the box should have the dimensions (2, 2, 1)

And minimum surface Area Amin = x^2 + 4xh = 2^2 + 4*2*1 = 4 + 8 = 12 in^2

Solution(C):

First you must come up with an equation of the yield.

Yield = Bushels per tree * number of trees

Bushels per tree = 44 when the number of trees is 24 and decreases by 1 for each additional tree.

x = number of trees

Bushels per tree = 44 when x = 24 so bushels per tree = (68 - x) = (68 - 24) = 44

Yield = (68 - x) * x

Yield = 68x - x2

For highest take the derivative and equal to zero.

=> 68 - 2x = 0

=> x = 68/2 = 34

Maximum yield will be given with 34 trees.

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