S.O.S. Mathematics CyberBoard

[orgchemus]

(1) American Airlines requires that the total outside dimensions (Length + Width + Height) of your carry on bag not exceed 63 inches.

Suppose you want to buy a carry on bag whose length is twice its height (L = 2H).

What does the height have to be so that the volume of the bag will be maximized?

(2) Hercules Films is deciding on the price of the video release of its film Bride of Frankenstein. The marketing department estimates that at a selling price of p dollars it can sell q = 400000 - 12000 p copies.

Each copy cost $4 to make.

What price p will give the greatest profit?

(3) There are 90 apple trees in an orchard. Each tree produces 1500 apples.

There is a limited amount of water that is used for irrigation. Because of this, for each additional tree planted in the orchard, the output per tree drops by 12 apples.

The owner of the orchard is considering planting more trees, but is concerned that if she plants too many, that overall production will drop.

How many trees should be added to the existing orchard in order to maximize the total output of trees ?

Production = Number of Trees * Output Per Tree

Let x = The Number of Trees that will be Added to the Orchard

Number of Trees =( 90 + x )

Output Per Tree = ( 1500 - 12x )

I may understand this problem and I believe it goes as follows:

(90+x)(1500-12x) = 135,000+420x-12x^2 =f(x)

f1(x)=-24x+420

0=-24x+420

24x=420

x=17.5 or 18 trees

[aswoods]

For question 3, your equations are correct, but you will have to justify the steps taken a bit better. You've found a stationary point, but you have to point out that it is a maximum, etc. The answer is 17 or 18, as the parabola is symmetrical about x=17.5.

The other questions are at the same level and I can't see why you should have any difficulties. In the first, L+W+H=63 and L=2H (they explicitly give this to you) and you need to maximize LWH, so find L and W in terms of H and substitute. In the second, the profit will be (number sold) * (price per unit - cost per unit).