More applications of a quadratic function

Example

The length and width of a rectangle are (3x + 1) and (2x – 1) cm respectively. If the area of the rectangle is 144 cm2, find x.

Identify the unknown!

(3 1)(2 1) 144x x Form the equation!

2

2

6 3 2 1 144

6 145 0

(6 29)( 5) 0

295 or

6

x x x

x x

x x

x

Solve!

Are both answers acceptable?: 5 Ans x

(rej)

ExampleA rectangular swimming pool measures 25 m by 6 m. It is surrounded by a path of uniform width. If the area of the path is 102 m2, find the width of the path.

Let the width be x. Therefore, length of path = 25 + 2x, breadth of path = 6 + 2x

(25 2 )(6 2 ) 252x x 2

2

2

150 50 12 4 252

4 62 102 0

2 31 51 0

( 17)(2 3) 0

1.5 or -17(rej)

x x x

x x

x x

x x

x

25 m6 m

25 + 2x

6 + 2x

Area of pool = 25 x 6 = 150 m2

Ans: The width of the path is 1.5 m

Example – Remember Abigail?Abigail, who has a bionic arm, is crossing a

bridge over a small gorge and decides to toss a coin into the stream below for luck. The distance of the coin above the water can be modeled by the function y= -16x2+96x+112, where x measures time in seconds and y measures the height, in feet, above the water.

Find the time at which the coin hits the water.

ExampleJack and Jill have a lemonade

stand in a really good location and want to increase their prices. They figure out the equation that models their business profit is P(x)=-2x2+11x-12.

What would you do to find their maximum profit???

At what price per cup would their profit be zero?