**Setting Up Functions**

In the following problems, we will be asked to set up a function of one variable to describe a certain quantity. In order to do this, we can

1. First set up a function of two variables for the desired quantity, using a picture if applicable.

2. Use the information given in the problem to find an equation relating the two variables in step 1, and then solve for one of the variables in terms of the other.

3. Substitute back into the function found in step 1 to get a
function of just one variable.

**Ex 1** A rectangle with base and height has a perimeter of 120 ft.

Express the area A of the rectangle as a function of .

**Sol**

1. We know that for a rectangle.

2. , so and therefore .

3. Substituting back gives .

**Ex 2** A right triangle has base and height . If the area of the
triangle is 5 square inches, express the perimeter of the triangle as a
function of .

**Sol**

1. Using the Pythagorean Theorem, we know that .

2. Since , and .

3. Substituting back gives

.

**Pr 1** A rectangle has base and height . If the area of the
rectangle is 50 square meters, express the perimeter of the rectangle
as a function of .

**Pr 2** A rectangular pasture has base and height
, and the bottom
side of the pasture is bounded by a river. If the other 3 sides of the
pasture
are bounded by 220 feet of fencing, express the area of the rectangle
as a
function of .

**Pr 3** A rectangle with base and height is inscribed in a circle of radius 5 inches. Express the
area of the rectangle as a function of .

**Pr 4** A rectangle has its base on the x-axis and its top two
vertices on
the graph of . If the top right vertex has
coordinates ,
Express the area of the rectangle as a function of .

**Pr 5** A right triangle has vertices at the origin and at the
points
and (with and positive). If the hypotenuse of the triangle passes
through the point , express the area of the triangle
as a function of .

**Pr 6** An athletic field has the shape of a rectangular region
with semicircular regions at each end. (See the diagram below.)

If the perimeter of the field is 400 meters, express the area of the field as a function of the radius of the semicircles.

**Pr 7** A rectangular page in a book has margins of 1.5 inch at
the top and the bottom and 1 inch on the right and the left. If the
printed material on the page is a rectangle with base and height , and its area is 50 square
inches, express the area of the page as a function of .

**Pr 8** A rectangular box has a square base and a volume of 80
cubic inches.
If the side of the base has length and the height of
the box is ,
express the surface area of the box as a function of .

**Pr 9** A rectangular box has a square base and a volume of 60
cubic inches. If the side of the base has length and the
height of the box is , and the
material for the top and bottom costs 10 cents per square inch and the
material
for the other four sides costs 8 cents per square inch, express the
cost of the
box as a function of .

**Pr 10** A right circular cylinder with height and base radius has
a volume of cubic inches. Express the total
surface area of the cylinder as
a function of . (Recall that the area of the side of the
cylinder is given
by .)

**Pr 11** A right circular cylinder with height and base radius has
a volume of cubic inches. If the material for
the top and the bottom of the
cylinder costs 12 cents per square inch, and the material for the side
of the
cylinder costs 9 cents per square inch, express the cost of the
cylinder as a
function of .

**Pr 12** A right circular cylinder with height and base radius has
a total surface area of square inches.
Express the volume of the
cylinder as a function of r.

**Pr 13** A ladder reaches over a fence 6 feet high to a wall 4
feet behind
the fence. Express the length of the ladder as a function of the
distance
from the base of the ladder to the fence.

**Pr 14** A hiker in the desert is 4 miles from a straight road,
and he wants
to walk to a town 10 miles down the road
from the point on the road closest to him. (See the diagram below.)

If he can walk 5 mph along the road and 3 mph off the road, express the time it will take him to reach the town as a function of the distance .

Go to Solutions.

Return to Precalculus Home Page.