This page contains a number of worked word problems. Of course, no list could be complete, but there are certain pattern that occur over and over.
Solution:
Let's say your previous wage was dollars per hour. You are now
making 10 percent more. In other
words, you are making
% of your old wage. Hence
Solution: The discount
applies to the list price of . The sales tax applies
to the discounted price.
% of
is
. The discounted price therefore is
(Equivalently you could argue that you buy
the pants for
% of the list price, which is
) You still need to pay
% sales tax
on the price of
. Since
% percent of
is
your total price is
Solution: Let's call the growth
rate . Every year the population gets multiplied with
. The problem tells us that
Solution: The main difficulty here is to
understand the meaning of the phrase 5% vinegar. Think of ordinary
vinegar as a mixture of water and pure (100%) vinegar. 5% vinegar
means that the ratio of the pure vinegar in the mixture, and the total
amount of fluid is 0.05 (or 5%). In this example the total amount of
the fluid is 3 gallons, and the total amount of pure vinegar is
gallons (i.e., 5% of the 1 gallon and 20% of the two
gallons). The ratio of pure vinegar to total amount of fluid is,
therefore,
Solution: The trick for this kind of
problem is to think of the job as the basic unit. You want to do 1
job, i.e., paint that room. You can do 1/2 of that job in one hour,
and your brother can do 1/3 of the job in one hour. Painting together
the two of you will do of that job in
hours. Thus we need to solve the equation
Solution: The solution to all such problems proceeds by drawing a picture, and then applying the Pythagorean Theorem. In this case we get the following Figure which shows the mountain in a fetching green color:
Denoting the unknown height by we get the equation
Solution: There are quite a
few problems like this in the homework sets because they provide
excellent exercises in applying radicals. As pointed out on the
scaling
page, multiplying each linear dimension with a factor multiplies
every area with the square of that factor, and the volume (and weight)
with the cube of that factor. Everything flows from that basic fact!
In this case we multiply linear dimensions with 3, and so we multiply
the area (and hence the price of the gold leaf) with and the
volume (and hence the weight) of the statue with
. You pay $1,800
for the gold leaf, and your original statue weights 54 tons.
You now begin to wonder how much it will cost to reinforce your floor, or whether you should perhaps remove the floor altogether and place your statue in the basement. But it is a beautiful statue!
In some of the homework problems in this course you are given
(explicitly or implicitly) a factor that multiplies the area or the
volume. As explained on the
scaling
page, if you multiply the areas with you multiply the
lengths with
and the volume with
, and
if you multiply the volume with
you multiply the lengths with
and the areas with
.
You throw a rock and release it at a height of feet. Its
initial vertical velocity is
feet per second (going up)
and its horizontal (forward) velocity is
feet per second.
How far does the rock fly until it hits the ground, and what is the
maximum height that it reaches?
Solution:
According to the discussion of
throwing
rocks the height of the rock at the time is given in general
by
In our case
feet per
second squared (since we are on earth),
feet per second
(the initial upward motion), and
feet, the height at which
we release the rock. The height of the rock at time
, therefore,
is given by
To find out at what time the
rock hits the ground (where the height is zero) we need to solve the
quadratic
equation
which gives
seconds as the time
of impact. During that time the rock covers a distance that's
obtained by multiplying the flight time with the horizontal velocity,
which gives the distance
feet. At the time
that the rock reaches its maximum height its vertical velocity, which
in general is given by
is zero. In our case
and hence the time
at which the rock reaches the
maximum height is given by
seconds. At
that time the rock is a height
feet.