Chapter Twenty-Three — Word Problems
The Humongous Book of Algebra Problems
525
The total interest accrued when compounded annually is
$11,766.49 – $3,500 = $8,266.49. Subtract this amount from the total interest
earned when compounded continuously (calculated earlier in the problem).
$8,716.20 – $8,266.49 = $449.71
The principal earns $449.71 more interest in a continuously compounding
account than it does in an account that compounds interest annually.
Geometric Formulas
Area, volume, perimeter, and so on
23.16 Calculate the radius of a circle that has a circumference of 10 inches.
The formula for the circumference C of a circle with radius r is C = 2pr.
Substitute C = 10 into the formula and solve for r.
The radius of the circle is inches.
23.17 Calculate the area of a square that has a perimeter of 50 cm.
The formula for the perimeter of a square with side length s is P = 4s. Substitute
P = 50 into the formula and solve for s.
The perimeter
is the sum of the
sides of a geometric
gure. A square has
four equal sides, so to
calculate its perimeter,
multiply the length of
one side by 4.
Chapter Twenty-Three — Word Problems
The Humongous Book of Algebra Problems
526
Each side of the square is inches long. Substitute into the
formula for the area of a square: A = s
2
.
The area of the square is
in
2
.
23.18 Calculate the area of a circle that has a diameter 8 cm long.
The radius of a circle is equal to half of its diameter: 8 ÷ 2 = 4. Substitute r = 4
into the formula for the area of a circle: A = pr
2
.
A = p(4)
2
= 16p
The area of the circle is 16p cm
2
.
23.19 If the base of a triangle is one unit longer than its height, and the area of the
triangle is 10 in
2
, what is the length of the base?
Let h represent the height of a triangle and b represent the base. The area of the
corresponding triangle is . Note that the height of the triangle is one
unit shorter than the base: h = b – 1. Substitute A = 10 and h = b – 1 into the
formula.
The area of
any rectangle
(including a
square, which is
technically also a
rectangle) is equal
to its length times its
width. The length
and width of a
square are equal,
so the area is the
length of one
side squared.
You can write
the answer in
decimal form:
(12.5)
2
= 156.25 in
2
.
If the length of the
square’s side is measured
in inches, then the area
is measured in square
inches (in
2
).
Make sure to
include units in your
nal answers. The
length is measured in
centimeters, so the
area is measured in
square centimeters
(cm
2
).
You could also set b = h + 1
and plug that into the formula.
You’ll end up calculating the
height. Plug that height into
b = h + 1 (in other words, add 1) to
calculate the length of the base.
Multiply
both sides of
the equation by
2 to eliminate
fractions.
Chapter Twenty-Three — Word Problems
The Humongous Book of Algebra Problems
527
Set the left side of the quadratic equation equal to zero and solve by factoring.
The sides of a triangle, like the sides of any geometric figure, must have positive
lengths, so discard the solution b = –4. The base of the triangle is 5 in long.
23.20 If the perimeter of a rectangle is 28 cm and the length of the rectangle is one
less than twice the width, what is the area of the rectangle?
The perimeter of a rectangle with length l and width w is P = 2l + 2w; the area is
A = lw. The length of this rectangle is one less that twice the width, so l = 2w – 1.
Substitute l and P into the perimeter formula and solve for w.
Substitute w = 5 into the length expression.
Substitute w = 5 and l = 9 into the area formula
A = lw = (9)(5) = 45
The area of the rectangle is 45 cm
2
.
The oppo-
site sides of a
rectangle are equal,
so there are two sides
L cm long and two sides
W cm long. Calculate
the perimeter by
adding up the lengths
of all the sides:
P = L + L + W + W =
2L + 2W.
Chapter Twenty-Three — Word Problems
The Humongous Book of Algebra Problems
528
23.21 Calculate the radius (in inches) of a right circular cylinder that is one foot tall
and has a volume of 96p in
3
.
The volume of a right circular cylinder with radius r and height h is V = pr
2
h.
Substitute V = 96p and h = 12 into the formula and solve for r.
The radius, like any geometric length, must be a positive number, so discard the
solution . The radius of the cylinder is inches.
23.22 Calculate the dimensions of a right rectangular prism with volume 60 in
3
if it
has a height of 5 in and its length is three more than four times its width.
The volume of a right rectangular prism with length l, width w, and height h
is V = lwh. Note that the length of this prism is three more than four times the
width, so l = 4w + 3. Substitute V = 60, h = 5, and l = 4w + 3 into the formula to
solve for w.
Divide the entire equation by 5, the greatest common factor.
Apply the quadratic formula to solve the equation.
The answer’s
supposed to be
in terms of inches,
but the height is
reported in feet.
Multiply h by 12 to
convert 1 foot into
12 inches.
“Right
rectangular
prism” is a fancy
way of saying
“box.”
Dividing
every term
of an equation
by the same
nonzero number
doesn’t affect the
solution, and the
smaller coefcients
you get as a result
make solving the
equation a little
easier.
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