Problems

1            Find the perimeter p of a square as a function of its area A.

2            A piece of clay in the shape of a cube of side s is rolled into a sphere of radius r. Find r as a function of s.

3            Find the volume V of a sphere as a function of its surface area S.

4            Find the area A of a rectangle of perimeter 4 as a function of the length x.

5            Find the distance z between the origin and a point on the parabola y = 1 - x² as a function of x.

6            Express the perimeter p of a right triangle as a function of the base x and height y.

7            Four small squares of side x are cut from the corners of a large cardboard square of side s. The sides are then folded up to form an open top box. Find the volume of the box as a function of s and x.

8            A ladder of length L is propped up against a wall with its bottom at distance x from the wall. Find the height y of the top of the ladder as a function of x.

9            A man of height y stands 3 ft from a ten foot high lamp. Find the length s of his shadow as a function of y.

10            One ship traveling north at 30 mph passes the origin at time t = 0 hours. A second ship moving east at 30 mph passes the origin at t = 1. Find the distance z between them as a function of t.

11             A ball is thrown from ground level, and its path follows the equations
x = bt, y = t - 16t².
How far does it travel in the x direction before it hits the ground?

12            A circular weedpatch is initially 2 ft in radius. It grows so that its radius increases by 1 ft/ day. Find its area after five days.

13            A rectangle originally has length l and width w. Its shape changes so that its length increases by one unit per second while its width decreases by 2 units per second. Find its area as a function of l, w and time t.

14            At p units of pollution per item, a product can be made at a cost of 2 + 1/p dollars per item, x items are to be produced with a total pollution of one unit. Find the cost.

15            In economics, the profit in producing and selling x items is equal to the revenue minus the cost,
P(x) = R(x) - C(x).

If a product can be manufactured at a cost of $10 per item and x items can be sold at a price of 100 - √x per item, find the profit as a function of x.

16            Suppose the demand for a commodity at price p is x = 1000/√p, that is, x = 1000/√p items can be sold at a price of p dollars per item. If it costs 100 + 10x dollars to produce x items, find the profit as a function of the selling price p.