**Word Problems: Maximum and Minimum**

Trajectory

1. A toy rocket is launched off the roof of a building. Its height h in metres at time t in seconds is given by the function h(t) = -4.9t^2 + 34.3t + 39.2

a) How high is the roof?

b) When will the rocket hit the ground?

c) When does the rocket pass at roof level on the way down?

d) When does the rocket achieve its maximum height?

e) What is the maximum height?

f) For how long is the rocket above 88.2 m?

ANSWERS: a) 39.2 m b) 8 seconds c) 7 seconds d) 3.5 seconds e) 99.2 m f) 3 seconds Video Solution

Revenue

2. Felix owns a business selling exotic snowglobes. He knows that if he sells them for $30 each, he can sell a total of 1600 snowglobes. However, for every one dollar increase in price, he sells 20 fewer snow globes.

a) What price should he charge per snow globe to maximize his revenue?

b) What is the maximum possible revenue?

ANSWERS: a) $55 b) $60 500 Video solution to question

Revenue and Profit (Orchard Owner)

3. An orchard owner is growing apple trees on an acre of property. If she grows 50 trees on the acre, she'll get 800 apples to grow on each tree. For every additional tree that she plants, there will be ten fewer apples grown on each tree. Furthermore, she can sell each apple for 25 cents, and it costs her 5 dollars to maintain each tree.

a) How many trees should she grow for the maximum possible yield of apples?

b) What is the maximum possible yield of apples?

c) What is the maximum possible revenue?

d) How many trees should she grow for the maximum possible profit?

e) What is the maximum possible profit?

ANSWERS: a) 65 trees b) 42 250 apples c) $10 562.50 d) 64 trees e) $10 240 Video Solution

Maximum Area (e.g., Fence around a Garden, Max. Sunlight)

4. Mohammed has 20 m of fence as shown. He wishes to put fencing around the outside of his garden. He plans to build it against his house so that there is no need for fencing to be put along that side.

a) What are the dimensions of the largest garden possible (maximum area)?

b) What is the area of the largest possible garden?

ANSWERS: a) 5 m by 10 m b) 50 m^2 Video Solution

5. Fatima has 72 m of fence as shown. She wishes to put fencing around the outside of her garden. She plans to build it against her house so there is no need for fencing to be put along that side.

a) What is the largest possible total area?

b) What are the possible dimensions if Fatima want each of the three pens to be 96 m^2 in area?

ANSWERS: a) 324 m^2 b) either each of the four identical sides is 6 m and the long side is 48 m, or each of the four identical sides is 12 m and the long side is 24 m. Video Solution to

6. Hemza has 60 metres of fence with which to build the rectangular garden shown. He plans to build it aginst his house so there is no need for fencing to be placed along that side. What is the maximum possible area that can be enclosed?

ANSWER: 150 m^2 Video Solution

7. A giant window is to be built using 1200 metres of steel. The window will be in the shape of an equilateral triangle over a rectangle.

a) What dimensions will let the most light through?

b) Suppose that the triangle is to be tinted so that only half as much light per unit area shines through. What dimensions will let the most light shine through?

* - You many round your answers to the nearest tenth of a metre.

ANSWERS: a) triangle sides approx 191.5 m, other rectangle sides approx 217.1 m; b) triangle sides approx 168.2 m, others 263.6 m

Video solution

8. A cow weighs 300 kg. The cow gains 6 kg per day. Beef sells at $2.70 per kg, and the price decreases $0.03 per day. It costs $5.40 per day to feed and maintain the cow.

a) When should the cow be sold to maximize profit?

b) How much will the cow weigh at that time?

c) What will the maximum profit be?

Answers: a) 5 days b) 330 kg c) $814.50 Video Solution

**Recap of Quadratic Inverses**

If you would like to review completing the square and partial factoring, please click here

__Radicals__

9. Simplify each of the following

a) SQRT(150)

b) 6 SQRT(3) - SQRT(48)

c) [ 9 + SQRT(3) ] [ 9 - SQRT(3) ]

d) [ SQRT(8) + SQRT(6) ] [ SQRT(6) - SQRT(2) ]

ANSWERS: a) 5 SQRT(6) b) 2 SQRT(3) c) 78 d) 2 + 2 SQRT(3) Video solution

__Determining the Range of a Quadratic over a Restricted Domain__

10. Determine the range of the parabola y = 3 (x-1)^2 + 5 over the domain -1 < x <= 2

Answer: {y/ 5 <= y < 17, y E R} Video Solution

__Word Problems: Finding Zeros__

Border Questions

11. __ Border Outside the Established Rectangle__: Sarah has a rectangular photo that is 30 cm X 40 cm. She want to build a frame of uniform width around the outside of it and she can afford 624 cm^2 of wood for the frame. How wide will the frame be?

ANSWER: 4 cm Video Solution

12. ** Border Inside the Established Rectangle**: Zheng's lawn is a rectangle measuring 20 m by 12 m. He wants to dig up some of his lawn in order to make a sidewalk of uniform width all around. How wide will the walkway be, given that he can afford 60 m^2 of cement?

ANSWER: 1 m Video Solution

Mixed Questions

13. A rectangle has one side 5 cm longer than the other. The total area is 176 cm^2. What are the dimensions of the rectangle?

ANSWER: 16 cm X 11 cm Video Solution

14. A rectangle has an area of 475 m^2. One side is 6m longer than the other side. What are the dimensions of the rectangle?

ANSWER: 19m and 25 m Video Solution

15. The sum of the squares of two consecutive integers is 925. What are the integers?

ANSWER: Either 21 and 22 or -22 and -21 Video Solution

16. The sum of the squares of three consecutive odd positive integers is 875. What are the integers?

ANSWER: 15, 17 and 19 Video Solution

17. A right angled triangle has a perimeter of 120 cm. The hypotenuse is 6 m longer than one of the sides. How long are the sides?

ANSWER: 24 cm, 45 cm and 51 cm. Video Solution

__Number of Zeros (Discriminant)__

18. Determine the number of real zeros in each of the following cases:

a) f(x) = 19x^2 + 27x + 3

b) f(x) = 3x^2 - 24x + 48

c) f(x) = -x^2 - 6x - 11

ANSWERS: a) 2 b) 1 c) 0 Video solution

19. For what value(s) of k will the function

a) f(x) = 2x^2 + 12x + k have exactly one real root?

b) g(x) = 3x^2 + kx + 192 have two distinct real roots?

c) h(x) = kx^2 - 6x +3 have zero real roots?

d) f(x) = kx^2 + 12x + k have two real roots?

ANSWER: a) k = 18 b) { k / k < -48, k > 48, k E **R** } c) { k / k > 3, k E **R** } d) { k / -6 < k < 6, k E **R** } Video solution

__Points of Intersection__

20. Determine the point(s) of intersection of the parabola y = -4x^2 + 7x + 32 and the straight line 5x + y + 8 = 0

ANSWER: (-2, 2) and (5, -33) Video solution

21. Determine the value(s) of k such that the linear function f(x) = 5x + k intersects the parabola h(x) = 3x^2 - 7x two times.

ANSWER: { k / k > -12, k E **R** } Video Solution

__Determining Equation of a Quadratic Function__

22. Determine the equation of a parabola in standard form with roots at x = -3 and x = 4, that passes through the point (1,60).

ANSWER: y = -5x^2 + 5x + 60 Video Solution

23. Determine the equation of a parabola in standard form with a vertex at (-3, 5) and that passes through the point (-5, 37)

ANSWER: y = 8x^2 + 48x + 77 Video Solution

24. Determine the equation in standard form of a parabola with x-intercepts at x = 4 + SQRT(5) and at x = 4 - SQRT(5) , and that passes through (7, 3).

ANSWER: y = 3/4 x^2 - 6x + 33/4 Video Solution

Word Problem

25. A tunnel with a parabolic arch is 8m wide. If the height of the arch 1 metre from the outer edge is 3 metres, can a truck that is 6 metres tall and 3 metres wide fit through the tunnel?

Answer: No Video Solution