Section 2.4C: Using Functions in Word Problems
Useful Geometry Results
You will NOT be provided these formulas on a test, so you should memorize most of them.
(Put more priority on the 2D formulas; spheres should be less important. Many of these formulas should
make sense with a picture!)
2D Formulas
CIRCLES (radius r)
Circumference = 2πr
Area = πr2
RECTANGLES (sides L and W )
Perimeter = 2L + 2W
Area = LW
TRIANGLES (base b, height h)
Perimeter = add up sides Area = (1/2)bh
PYTHAGORAS (legs a and b, hyp c) Right triangles:
a2 + b2 = c2
3D Formulas
SPHERES (radius r)
Surface Area = 4πr2
Volume = (4/3)πr3
BOX (sides L, W , H)
Surface Area = add six face areas Volume = LW H
CYLINDER (radius r, height h) Surface Area = 2πr2 + 2πrh
Volume = πr2 h
CONE (radius r, height h)
Volume = (1/3)πr2 h
Miscellaneous
SPEED speed = distance / time
OR: distance = (speed)*(time)
TOTAL COST Cost = (price per item) * (quantity)
Notes about the cylinder: Volume is (area of base) * (height), and the base is a circle with area πr2 . For the
surface area, you have two lids which each have area πr2 , and then you have the “siding” of the cylinder whose
area is (circum) * (height).
Approach to Word Problems
Here are some guidelines for breaking up a word problem into steps:
1. PICTURE: Draw a simple sketch.
2. VARIABLES: Which variables are you going to use? What is your goal? Which variables are allowed in
your answer? (Look for phrases like “in terms of” or “as a function of”.)
3. FORMULAS: Put down equations that use your unknowns. Plug in the values you’re given.
4. ISOLATE: Get your goal by itself in an equation. Note which variables you don’t want.
5. ELIMINATE: Use the other equations to eliminate the unwanted variables.
Warmup: Two numbers multiply to 40. If x is one of the numbers, write the sum of their square roots as
a function of x.
• Variables: x and y, the two numbers.
√
√
• Formulas: We’re told xy = 40. We want x + y.
• Isolate: We want to eliminate y, so use xy = 40 to get y = 40/x.
r
√
√
40
√
• Eliminate: x + y becomes
x+
.
x
Several 2D Problems
1. A rectangle has perimeter 30. Find its area A
in terms of its base length x.
NOTE: How would the problem change if you were
instead given area and wanted perimeter?
2. The point P (x, y) lies on the graph of y = 2x4
(see the picture on the right). Find the perimeter of
the right triangle as a function of x.
3. Express the height h of an equilateral triangle as a function of the side length x.
NOTE: This problem involves a useful skill: breaking up a shape into simpler shapes! (For instance, the
original triangle is not a right triangle, but we draw two right triangles inside it.)
4. A small office unit is to contain 900 square feet
of floor space. (See the figure to the right.)
(a) Find the length y of the building in terms of x.
(b) If walls cost $200 per foot, find the total cost C
of the walls as a function of x.
NOTE: In 4(b), you find a length by subtracting parts you don’t want! This “subtractive” reasoning can be
very helpful sometimes.
Several 3D Problems
5. From a rectangular a × b piece of cardboard,
where a = 40 inches and b = 50 inches, an open box is
made by cutting out a square of area x2 from each corner and turning up the sides (see the figure). Express
the volume V of the box as a function of x.
6. A right circular cylinder with radius r and height h. The volume of the cylinder is 4600 cubic feet.
The bottom of the cylinder is reinforced steel, costing $55 per square foot, whereas the sides and top cost
just $5 per square foot. Express the total cost of the cylinder as a function of r.
NOTE: The key idea here is that for each part, cost = (price per area)*(area). Since the prices aren’t all
the same, we want to treat the different parts of surface area separately.
Miscellaneous
7. A hot-air balloon is released at 1:00 P.M. and
rises vertically at a rate of 6 m/sec. An observation
point is situated 100 meters from a point on the ground
directly below the balloon (see the figure). If t denotes
the time (in seconds) after 1:00 P.M., express the distance d between the balloon and the observation point
as a function of t.
NOTE: In this case, the “1:00PM” data is actually unneeded in the calculations!
8. Two cars leave an intersection at the same time. Car A travels north at 50 miles per hour, and Car B
travels east at 40 miles per hour. Find the distance (in miles) between the two cars at time t, where t represents
the number of minutes elapsed since the cars left the intersection.
Write your answer as an expression in terms of the variable t.
NOTE: In this problem, we have to be careful about units! The speeds use hours, but t is in MINUTES, so
we multiply our speeds by a conversion factor. For instance,
50 mph =
5
50 miles 1 hour
·
= miles / min
1 hour 60 min
6
Lessons to Keep in Mind
• Usually, one equation gives you your goal. It will tell you what you need to eliminate! Use the other
equations to help you do that.
• Read over the problem to make sure you used ALL the data.
– Once again, reread to make sure you’re answering what is asked! For instance, if the question asks
for volume, do not give surface area as your answer.
– Rarely, you’ll want to convert units. Double-check that your units match up.
• Break tricky shapes into simple shapes. (This leads to adding or subtracting the values of several parts.)
• DO NOT go about studying these by saying “Oh, I need to study that problem with the two cars!” Ask
yourself instead, “Which parts of the question or picture told me which formula to use?”
See the schedule on my course site for some extra outlines of problems we didn’t solve today!