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

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

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.

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

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

Now You Try -1

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

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

Check the Answers x = -½ or x = 3

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

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

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

Now You Try -2, 0

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 + 17.5 Mixture = x + 50

Mixture Problem

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

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

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

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

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

Finding a Minimum Perimeter [-1, 40] x [-1, 300] The sides should be 15.8 ft and 31.6 ft for a perimeter of 63.25 ft.

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

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.

What We Need To Know r = radius of can h = height of can 1 L = 1000 cm3 V = r2h SA = 2r2 + 2rh

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 29.83 cm or have a radius of 9.65 and a height of 6.83 cm.

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 = 1.12 h = 126.88 r = 11.37 h = 1.23

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