6.6 Solving Problems Using Quadratic Models
Chapter
Chapter 6
Section
6.6
Solutions 34 Videos

For the relation, explain what each coordinate of the vertex represents and what the zeros represent.

• a relation that models the height, h, of a ball that has been kicked from the ground after time t
1.14mins
Q1a

For the relation, explain what each coordinate of the vertex represents and what the zeros represent.

• a relation that models the height, h, of a ball when it is a distance of x metres from where it was thrown from a second-floor balcony
0.51mins
Q1b

For the relation, explain what each coordinate of the vertex represents and what the zeros represent.

• a relation that models the profit earned, P, on an item at a given selling price, s
0.44mins
Q1c

For the relation, explain what each coordinate of the vertex represents and what the zeros represent.

• a relation that models the cost, C, to create n items using a piece of machinery
1.13mins
Q1d

For the relation, explain what each coordinate of the vertex represents and what the zeros represent.

• a relation that models the height, h, of a swing above the ground during one swing, t seconds after the swing begins to move forward
0.32mins
Q1e

A model rocket is shot straight up from the roof of a school. The height, h, in metres, after t seconds can be approximated by h=15+22t-5t^2.

• What is the height of the school?
0.27mins
Q2a

A model rocket is shot straight up from the roof of a school. The height, h, in metres, after t seconds can be approximated by h=15+22t-5t^2.

• How long does it take for the rocket to pass a window that is 10 m above the ground?
1.35mins
Q2b

A model rocket is shot straight up from the roof of a school. The height, h, in metres, after t seconds can be approximated by h=15+22t-5t^2.

• When does the rocket hit the ground?
0.53mins
Q2c

A model rocket is shot straight up from the roof of a school. The height, h, in metres, after t seconds can be approximated by h=15+22t-5t^2.

• What is the maximum height of the rocket?
1.06mins
Q2d

Water from a fire hose is sprayed on a fire that is coming from a window. The window is 15 m up the side of a wall. The equation H=-0.011x^2+0.99x+1.6 models the height of the jet of water, H, and the horizontal distance it can travel from the nozzle, x, both in metres.

• What is the maximum height that the water can reach?
0.53mins
Q3a

Water from a fire hose is sprayed on a fire that is coming from a window. The window is 15 m up the side of a wall. The equation H=-0.011x^2+0.99x+1.6 models the height of the jet of water, H, and the horizontal distance it can travel from the nozzle, x, both in metres.

• How far back could a fire-fighter stand, but still have the water reach the window?
2.08mins
Q3b

Brett is jumping on a trampoline in his backyard. Each jump takes about 2 s from beginning to end. He passes his bedroom window, which is 4 m high, 0.4 s into each jump. By modelling Brett's height with a quadratic relation, determine his maximum height.

2.41mins
Q4

Pauline wants to sell stainless steel water bottles as a school fundraiser. She knows that she will maximize profits of \$1024 if she sells the bottles for \$28 each. She also knows that she will lose \$4160 if she sells the bottles for only \$10 each.

• Write a quadratic relation to model her profit, P, in dollars, if she sells the bottles for x dollars each.
1.22mins
Q5a

Pauline wants to sell stainless steel water bottles as a school fundraiser. She knows that she will maximize profits, and raise \$1024, if she sells the bottles for \$28 each. She also knows that she will lose \$4160 if she sells the bottles for only \$10 each.

• What selling price will ensure that she breaks even?
0.50mins
Q5b

A professional stunt performer at a theme park dives off a tower, which is 21 m high, into water below. The performer's height, h, in metres, above the water at t seconds after starting the jump is given by h=-4.9t^2+21.

• How long does the performer take to reach the halfway point?
1.06mins
Q6a

A professional stunt performer at a theme park dives off a tower, which is 21 m high, into water below. The performer's height, h, in metres, above the water at t seconds after starting the jump is given by h=-4.9t^2+21.

• How long does the performer take to reach the water?
0.30mins
Q6b

A professional stunt performer at a theme park dives off a tower, which is 21 m high, into water below. The performer's height, h, in metres, above the water at t seconds after starting the jump is given by h=-4.9t^2+21.

• Compare the times for parts a) and b). Explain why the time at the bottom is not twice the time at the half-way point.
1.09mins
Q6c

Harold wants to build five identical pig pens, side by side, on his farm using 30 m of fencing. Determine the dimensions of the enclosure that would give his pigs the largest possible area. Calculate this area.

2.05mins
Q7

A biologist predicts that the deer population, P, in a certain national park can be modelled by P=8x^2-112x+570, where x is the number of years since 1999.

• According to this model, how many deer were in the park in 1999?
0.31mins
Q8a

A biologist predicts that the deer population, P, in a certain national park can be modelled by P=8x^2-112x+570, where x is the number of years since 1999.

• In which year was the deer population a minimum? How many deer were in the park when their population was a minimum?
1.41mins
Q8b

A biologist predicts that the deer population, P, in a certain national park can be modelled by P=8x^2-112x+570, where x is the number of years since 1999.

• Will the deer population ever reach zero, according to this model?
0.19mins
Q8c

A biologist predicts that the deer population, P, in a certain national park can be modelled by P=8x^2-112x+570, where x is the number of years since 1999.

• Would you use this model to predict the number of deer in the park in 2020? Explain.
0.55mins
Q8d

The depth underwater, d, in metres, of Daisy the dolphin during a dive can be modelled by d=0.1t^2-3.5t+6, where t is the time in seconds after the dolphin begins her descent toward the water.

• How long was Daisy underwater?
1.16mins
Q9a

The depth underwater, d, in metres, of Daisy the dolphin during a dive can be modelled by d=0.1t^2-3.5t+6, where t is the time in seconds after the dolphin begins her descent toward the water.

• How deep did Daisy dive?
0.51mins
Q9b

The cost, C, in dollars per hour, to run a machine can be modelled by C=0.01x^2-1.5x+93.25, where x is the number of items produced per hour.

• How many items should be produced each hour to minimize the cost?
0.51mins
Q10a

The cost, C, in dollars per hour, to run a machine can be modelled by C=0.01x^2-1.5x+93.25, where x is the number of items produced per hour.

• What production rate will keep the? cost below \$53?
1.37mins
Q10b

Nick has a beautiful rectangular garden, which measures 3 m by 3 m. He wants to create a uniform border of river rocks around three sides of his garden. If he wants the area of the border and the area of his garden to be equal, how wide should the border be?

2.37mins
Q11

A ball was thrown from the top of a playground jungle gym, which is 1.5 m high. The bill reached a maximum height of 4.2 m when it was 3 m from where it as thrown. How far from the jungle gym was the ball when it hit the ground?

3.09mins
Q12

The sum of the squares of three consecutive even integers is 980. Determine the integers.

1.50mins
Q13

Maggie can kick a football 34 m, reaching a maximum height of 16 m.

• Write an equation to model this situation.
1.37mins
Q14a

Maggie can kick a football 34 m, reaching a maximum height of 16 m.

• To score a field goal, the ball has to pass between the vertical poles and over the horizontal bar, which is 3.3 m above the ground. How far away from the uprights can Maggie be standing so that she has a chance to score a field goal?