Precalculus: Word Problems NOTES
1. A wire of length x is bent into the shape of a square.
a) Express the perimeter of the square as a function of x.
x
b) Express the area of the square as a function of x.
c) What if the wire had been curved into the shape of a circle? Find the function for perimeter and for area.
2. A circle of radius r is inscribed is a square (see the figure)
a) Express the area A of the square as a function of the radius r of the circle.
r
b) Express the perimeter p of the square as a function of r.
3. The volume of a right circular cone is V 
1 2
 r h . If the height is twice the radius, express the volume V as a function of the
3
radius.
4. a) A right triangle has one vertex on the graph of y  9  x at (x,y), another at the origin and the third on the positive
x – axis at (x, 0). See the figure. Express the area of the triangle as a function of x.
2
b) Using the parabola above, sketch three possible isosceles triangles where each has one vertex on the origin and the base has
endpoints on the curve. Draw the triangles only in QI and QII. Find the areas of each of the triangles.
If x = 1, area = __________
If x = 2, area = __________
If x = ____, area = _________
5. Building an enclosure along a river (or wall or barn.)
You have 500 feet of fencing. Maximize the enclosed area.
x
6. Walkway around a pool or garden. Shaded area = 32, find x.
x
8
x
x
x
12
7. You have 190 feet of fencing to make a rectangular region as shown. There is a 10ft x 15ft storage shed in the corner. What is the
max area (not including the shed) and what are the dimensions of the space? Include a drawing with your answer.
x
MORE NOTES
8. Constructing a closed box. A closed box with a square base is required to have a volume of 10 cubic feet.
a) Sketch a diagram for a 1 x 1 base, a 2 x 2 base, and a 5 x 5 base.
b) Express the amount of material (M) to make the
box as a function of the length of the square base (x).
c) Graph y = M(x). Whoa…hold up, we need a graphing calc for this!
Show your window and make a general sketch below.
d) For what values of x is the surface area the smallest?
What is the surface area?
9. Graph f  x  
x . Let P = (x,y) be a point on the graph of f  x  .
a) Express the distance (d) from P to the point (1, 0) as a function of x.
b) Sketch on the diagram where you think the distance will be the smallest.
c) Why would it be silly for me to ask where d is the largest?
d) Graph d(x) on your calc. Sketch below.
e) For what value(s) of x is d smallest?
10. Graph f  x   x  1 . Let P = (x,y) be a point on the graph of f  x  .
2
a) Express the distance (d) from P to the origin as a function of x.
b) What is the distance if x 
2
?
2
c)Sketch on the diagram where you think the distance will be the smallest.
d) Graph d(x) on your calc. Sketch below.
e) For what value(s) of x is d smallest?
11. A rectangle is inscribed in a semicircle of radius 2. Let P = (x,y) be the point in quadrant I that is the vertex of the rectangle and is on
the circle. See the figure.
a) Express the area A of the rectangle
as a function of x.
b) Express the perimeter p of the rectangle
as a function of x.
c) Graph A(x). For what value of x is A largest?
d) Graph p(x). For what value of x is p largest?
MORE Word Problems
12.
Getting from an Island to Town An island is 2 miles from the nearest point P on a straight shoreline. A town is 12 miles
down the shore from P.
a)
If a person can row a boat at an average speed of 3 miles per hour and the same person can walk 5 miles per hour,
express the time T that is takes to go from the island to town as a function of the distance x from P to where the
person lands the boat.
b)
What is the domain of T?
Lowest x (for the problem) _______
Highest reasonable x for the problem _______
Lowest x (in theory)
Highest x (in theory) ________
_______
Domain for the function _______________
Domain for this question _______________
Why are the domains different?
c)
e)
How long will it take to travel from
the island to town if the person goes directly?
Sketch this path.
Graph the function T=T(x) on your calculator.
Sketch below. Show the window.
d) How long will it take to travel from the island
to the town if you go straight to the shore and
then to the town? Sketch this path.
f) Use the CALC menu to find the value of x that results
in the least time. Sketch this path.
13. Susie has moved into a shanty in the woods with no plumbing. The local contractor informs her that it will cost $35/ft to lay piping
along the road and $50/ft to lay piping across her lawn. Her house is 100 ft from the closest point P on a straight road and the
nearest piping is 350 ft down the road from P
a. Express the total cost C of installation as a function of the distance x (in feet) from the nearest piping to the point where
the piping turns off the road. Give the domain.
b.
Compute the cost if x = 30 feet.
Sketch this situation.
c) Graph C(x).
d) What value of x results
in the least cost? What is
the least cost?
14. A manufacturer of playpens makes a square model that can be opened at one corner and attached to a wall. If each side is 3 feet in
length, the open configuration doubles the available play area from 9 square feet to 18 square feet. If we place hinges at the outer
corners (like the picture) we can increase the area again.
Top views of three configurations.
3
a)
Build a model that expresses the area A of the hinged configuration as a function of the distance
x between the two parallel sides.
b)
Find the domain of A(x) for this problem.
d)
For what value of x is the area the largest? What is the maximum area?
3
3
6
c) Find A if x = 5.
x