1. Mr. J. Focht Pre-Calculus OHHS
Chapter 2
Polynomial, Power, and
Rational Functions
Mr. J. Focht
Pre-Calculus
OHHS

2. 2.7 Solving Equations in One Variable
What We'll Learn
Solving Rational Equations
Extraneous Solutions
Applications

3. What You’ll Do
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 cm2.

4. Solving Rational Equations
Multiply by x to clear the fraction x x x
You may want to use the Quadratic Formula

5. Solving Rational Equations
Check your answers
x = 1 or x = 3

6. Now You Try -1

7. Solving Rational Equations
(x-1)(x-3)
(x-1)(x-3)
(x-1)(x-3)
(x-3)2x + x-1 = 2

8. Solving Rational Equations
(x-3)2x + x-1 = 2
2x2 – 6x + x-1 = 2
2x2 – 5x - 3 = 0
(2x + 1)(x – 3) = 0
x = -½ or x = 3

9. Check the Answers
x = -½ or x = 3

10. x = -½ or x = 3 Check the Answers
3 doesn’t work. It is an extraneous root.

11. Check Graphically Doesn’t cross the x-axis at 3
Crosses the x-axis at 0.5
[-5,5] x [-10, 10]

12. Solve algebraically. Confirm graphically.
Now You Try
Solve algebraically. Confirm graphically.

13. Now You Try
-2, 0

14. Mixture Problems
How much pure acid must be added to 50mL of a 35% acid solution to produce a mixture that is 75% acid?
x mL
Pure acid = x + 35%(50)
= x
Mixture = x + 50

15. Mixture Problem

16. Mixture Problems
Verify graphically
[0, 160] x [-1, 1]

17. Writing to Learn
You would add 82 mL of pure acid to the 50 mL of 35% solution to create a 75% solution.

18. Now You Try
Suppose that x mL of pure acid are added to 125 mL of a 60% acid solution. How many mL of pure acid must be added to obtain a solution of 83% acid? a) Find a function that finds the concentration of the new mixture. b) Write and solve the equation that answers the question.
169.12

19. Finding a Minimum Perimeter
Find the least amount of fencing if the area must be 500 ft2.
Only 3 sides are needed. The 4th side is a building.
500 ft2 x x

20. Finding a Minimum Perimeter
The perimeter is the function we want to minimize.

21. Finding a Minimum Perimeter
[-1, 40] x [-1, 300]
The sides should be 15.8 ft and 31.6 ft for a perimeter of ft.

22. Now You Try
Considering all rectangles with an area of 182 ft2. Let x be the length of one side of such a rectangle.
Express the perimeter P as a function of x.
Find the dimensions of the rectangle that has the least perimeter. What is the least perimeter?
53.96 ft

23. 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 cm2.

24. What We Need To Know r = radius of can h = height of can
1 L = 1000 cm V = r2h
SA = 2r2 + 2rh

25. Solving the Equation r=4.62 h = 29.63 r=9.65 h = 6.83
[2, 10] x [700, 1100]
r=9.65
h = 6.83
With a surface area of 1000 cm2, the cans either have a radius of 4.62 cm and a height of cm or have a radius of 9.65 and a height of 6.83 cm.

26. Now You Try
Drake Cannery will pack peaches in 0.5-L cylindrical cans. Let x be the radius of the can in cm.
Express the surface area S of the can as a function of x.
Find the radius and height of the can if the surface area is 900 cm2.
r = h =
r = h = 1.23

27. Home Work
P. 253 – 256
#2, 8, 12, 14, 18, 32, 38, 45-50