Examples
The area of a triangle is 40 in2.
Its height is four-fifths the length of its base. What are its base and height?
The area is 40 and H = ⅘B so the formula A = ½ BH becomes 40 = ½ B(⅘B).
The triangle’s base is 10 inches long and its
height is (⅘)(10) = 8 inches.
The hypotenuse of a right triangle is 34
feet. The sum of the lengths of the two legs is 46 feet. Find the lengths of
the legs.
The
sum of the lengths of the legs is 46 feet, so if a and b are the lengths
of the legs, a + b = 46, so a = 46 – b. The hypotenuse is 34 feet so if c is the length of the hypotenuse, then
the formula a2 + b2 = c2 becomes (46 – b)2
+ b2 = 342.
One leg is 30 feet long and the other is 46 –
30 = 16 feet long.
A can’s height is four inches and its volume
is 28 cubic inches. What is the can’s radius?
The
volume formula for a right circular cylinder is V = πr2h. The can’s
volume is 28 cubic inches and its height is 4 inches, so V = πr2h becomes 28 = πr2(4).
The can’s radius is about 1.493 inches.
The volume of a box is 72 cm3. Its
height is 3 cm. Its length is 1.5 times its width. What are the length and
width of the box?
The
formula for the volume of the box is V =
LWH. The volume is 72, the height is
3 and the length is 1.5 times the width (L
= 1.5W) so the formula becomes 72
= (1.5W)W(3).
The box’s width is 4 cm and its length is (1.5)(4)
= 6 cm.
The surface area of a ball is 314 square
inches. What is the ball’s diameter?
The
formula for the surface area of a sphere is SA
= 4 πr2. The area is 314, so the formula becomes 314
= 4 πr2.
The radius of the ball is approximately 5
inches. The diameter is twice the radius, so the diameter is approximately 10
inches.
The manufacturer of a six-inch drinking cup
is considering increasing its radius. The cup has straight sides (the top is
the same size as the bot- tom). If the radius is increased by one inch,
the new volume would be 169.6 cubic inches. What is the cup’s current radius?
The
formula for the volume of a right circular cylinder is V = πr2h. The cup’s height is 6. If the cup’s radius is increased, the volume
would be 169.6. Let x represent the
cup’s current radius. Then the radius of the new cup would be x + 1. The volume formula becomes 169.6 =
n(x + 1)26.
The cup’s current radius is approximately 2
inches.
Practice
1. The area
of a triangle is 12 in2. The length of its base is two-thirds its height. What
are the base and height?
2. The area
of a triangle is 20 cm2. The height is 3 cm more than its base. What are the
base and height?
3. The sum of the base and height of a triangle is 14 inches. The area is 20 in2. What are the base and height?
4. The
hypotenuse of a right triangle is 85 cm long. One leg is 71 cm longer than the
other. What are the lengths of its legs?
5. The
manufacturer of a food can wants to increase the capacity of one of its cans.
The can is 5 inches tall and its diameter is 6 inches. The manufacturer wants
to increase the can’s capacity by 50% and wants the can’s height to remain 5
inches. How much does the diameter need to increase?
6. A pizza
restaurant advertises that its large pizza is 20% larger than the competition’s
large pizza. The restaurant’s large pizza is 16 inches in diameter. What is the
diameter of the competition’s large pizza?
Solutions
1. The area
formula for a triangle is A = ½ BH. The area is 12. The length of its
base is two-thirds its height, so B =
⅔ H. The formula becomes 
.
The height of the triangle is 6
inches. Its base is (⅔)6 = 4
inches.
2. The
formula for the area of a triangle is A = ½ BH. The area is 20. The height is 3
cm more than the base, so H = B + 3. The formula becomes 20 = ½ B(B + 3).
The triangle’s base is 5 cm and its height is
5 + 3 = 8 cm.
3. The
formula for the area of a triangle is A = ½ BH. The area is 20. B + H = 14 so H
= 14 – B. The formula becomes 20 = ½ B(14 – B).
There are two triangles that satisfy the
conditions. If the base is 10 inches, the height is 14 – 10 = 4 inches. If the
base is 4 inches, the height is 14 – 4 = 10 inches.
4. By the
Pythagorean theorem, a2 + b2 = c2. The
hypotenuse is c, so c = 85. One leg is 71 longer than the other so a = b + 71 (b
= a + 71 also works). The Pythagorean theorem becomes 852 = (b + 71)2
+ b2.
The shorter leg is 13 cm and the
longer leg is 13 + 71 = 84 cm.
5. Because
the can’s diameter is 6, the radius is 3. Let x represent the increase in the
can’s radius. The radius of the new can is 3 + x. The volume of the current can
is V = πr2h = π(3)2
5 = 45π. To increase the volume by 50% means to add half of 45π to itself; the new volume would be 
. The volume formula for the new can becomes .png){kind=small}
.
(The other solution is negative.)
The manufacturer should increase the
can’s radius by about 0.674 inches. Because the diameter is twice the radius,
the manufacturer should increase the can’s diameter by about 2(0.674) = 1.348 inches.
6. A
pizza’s shape is circular so we need the area formula for a circle which is A =
πr2. The radius is
half the diameter, so the restaurant’s large pizza has a radius of 8 inches.
The area of the restaurant’s large pizza is π(8)2 = 64π ≈ 201.
The restaurant’s large pizza is 20% larger than the competition’s large pizza.
Let A represent the area of the competition’s large pizza. Then 201 is 20% more
than A:
The competition’s radius is approximately 7.3
inches, so its diameter is approximately 2(7.3) = 14.6 inches.
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