# Intermediate Algebra Worksheet

1. One often needs to multiply polynomials in problems involving areas and volumes.

(a) The length of a rectangular field is 10 meters more
than twice the width. Let w be the width.

What is the area of the field?

(b) A triangular roof panel has a height half as long as
its base. If the base has length b, what is the

area of the roof panel.

(c) A rectangular block has a width 5 meters shorter than
three times its length. The height is 3

times the length. If the width of the block is w, what is its volume?

(d) A silo is in the shape of a cylinder with a
hemispherical top. The radius of the silo is one third of

its height. If the height is h, find the volume of the silo.

2. Bill Gates goes insane and offers you that on the first
day of the month he will give you 1 dollar, on the

second day f dollars, and the third day f^{2} dollars, on the fourth day f^{3}
dollars, and so on, increasing

by a factor of f every day until the end of the month.

(a) The amount of money you have collected after d days is given by the formula

(1)

Verify that this formula works after 2 days, 4 days, and 6
days using f = 2, i.e. assuming that

the amount he gives you doubles every day.

(b) Use synthetic division to rewrite the formula for d =
4 days. Check that your result agrees with

your answer to part (a) for f = 2.

(c) Use synthetic diffision to rewrite the formula for d =
6. Again, check against your answer to part

(a).

3. A box has a length 10 cm less than twice its width. If
the width is w cm and the volume of the box is

2w^{3} − 20w^{2} + 50w cm^{3}, then:

(a) Use synthetic division to find the height of the box in terms of w.

(b) What is the area of the base of the box?

(c) What are the areas of the box sides?

(d) Find the the volume of a box whose with the same length and width, but twice the height.