Sec. 2.8b
Applications
Involving Rational
Equations
Calculating Acid Mixtures
How much pure acid must be added to 50 mL of a 35%
acid solution to produce a mixture that is 75% acid?
mL of pure acid
mL of mixture
= concentration of acid
mL of acid in 35% solution: (0.35)(50), or 17.5
mL of acid added: x
mL of pure acid in resulting mixture: x + 17.5
mL of the resulting mixture: x + 50
Calculating Acid Mixtures
How much pure acid must be added to 50 mL of a 35%
acid solution to produce a mixture that is 75% acid?
x + 17.5
x + 50
= concentration of acid
x + 17.5
= 0.75
x + 50
Let’s solve this graphically…
Point of intersection: (80, 0.75)
We need to add 80 mL of pure acid to the 35% acid
solution to make a solution that is 75% acid
Finding a Minimum Perimeter
Find the dimensions of the rectangle with minimum
perimeter if its area is 200 square meters. Find this least
perimeter.
x
200
x
 200 
A  200  x 

 x 
Finding a Minimum Perimeter
Find the dimensions of the rectangle with minimum
perimeter if its area is 200 square meters. Find this least
perimeter.
Perimeter = 2(length) + 2(width)
 200  Let’s minimize this function!!!
P  2x  2 

Calculator!!!
 x  Min. P of 56.569 meters at
x = 14.142 meters
400
P  2x 
Dimensions: 14.142 m by 14.142 m
x
Page 256, #36
The diagram:
0.75 in.
1.5 in.
(a) Area as a function of x:
 40

A  x    x  1.75    2.5 
 x

(b) Minimize this function (graph!):
40 1 in.
x
x
1 in.
Min. at
5.292,70.833
Dimensions of about 7.042 in. by
10.059 in. yield a minimum area
of about 70.833 square inches.
Page 256, #38
The diagram:
2
(a) Area as a function of x:
 1000

A  x    x  4 
 4
 x

1000 2
(b) Minimize this function (graph!):
x
2
x
2
Min. at
31.623,1268.982
 The pool is square!!!
With dimensions of approximately 35.623 ft x 35.623 ft,
the plot of land has minimum area of about 1268.982 sq ft.
Designing a Juice Can
Stewart Cannery will package tomato juice in 2-liter cylindrical
cans. Find the radius and height of the cans if the cans have
a surface area of 1000 square centimeters.
S = surface area of can (square centimeters)
r = radius of can (centimeters)
h = height of can (centimeters)
Note: 1 L = 1000 cubic centimeters
Designing a Juice Can
Stewart Cannery will package tomato juice in 2-liter cylindrical
cans. Find the radius and height of the cans if the cans have
a surface area of 1000 square centimeters.
V   r h  2000 S  2 r  2 rh  1000
2000
 2000 
2
h
2 r  2 r 
 1000
2

2
r
 r 
2
2
4000
2 r 
 1000
r
2
Designing a Juice Can
Stewart Cannery will package tomato juice in 2-liter cylindrical
cans. Find the radius and height of the cans if the cans have
a surface area of 1000 square centimeters.
4000
2 r 
 1000 Solve Graphically…
r
r = 4.619 cm, or r = 9.655 cm
2
Find the corresponding heights…
With a surface area of 1000 cm 2, the cans either have a
radius of 4.619 cm and a height of 29.839 cm, or have a
radius of 9.655 cm and a height of 6.829 cm.