(Grade 11) Section 3.5 - Solving Problems Involving Quadratics

1) Kevin threw a ball into the air. The height of the ball is modelled by the function h(t) = –5t²+35t, where h(t) is the height of the ball in metres and t is the time in seconds. When does the ball hit the ground?

1s

3.5s

7s

61.25s

The ball hits the ground when its height is equal to zero. To find the time when this occurs, we can set the height function h(t) equal to zero and solve for t. In this case, the equation -5t^2 + 35t = 0 represents the height of the ball. By factoring out a common factor of t, we get t(-5t + 35) = 0. This equation is satisfied when t = 0 or -5t + 35 = 0. Solving the second equation, we find t = 7. Therefore, the ball hits the ground at 7 seconds.

Explanation

The ball hits the ground when its height is equal to zero. To find the time when this occurs, we can set the height function h(t) equal to zero and solve for t. In this case, the equation -5t^2 + 35t = 0 represents the height of the ball. By factoring out a common factor of t, we get t(-5t + 35) = 0. This equation is satisfied when t = 0 or -5t + 35 = 0. Solving the second equation, we find t = 7. Therefore, the ball hits the ground at 7 seconds.

2) A seagull drops a sea urchin onto the rocks below from a height of 80m. The function h(t) = –5t²+80 models the height of the urchin, in metres, and t is the time in seconds. When will the urchin hit the rocks?

–4s

4s

5s

80s

The function h(t) = -5t^2 + 80 models the height of the urchin, where t is the time in seconds. To find when the urchin hits the rocks, we need to find the value of t when h(t) = 0. Setting -5t^2 + 80 = 0, we can solve for t. Rearranging the equation, we get 5t^2 = 80. Dividing both sides by 5, we have t^2 = 16. Taking the square root of both sides, we get t = ±4. Since time cannot be negative in this context, the urchin will hit the rocks after 4 seconds. Therefore, the correct answer is 4s.

Explanation

The function h(t) = -5t^2 + 80 models the height of the urchin, where t is the time in seconds. To find when the urchin hits the rocks, we need to find the value of t when h(t) = 0. Setting -5t^2 + 80 = 0, we can solve for t. Rearranging the equation, we get 5t^2 = 80. Dividing both sides by 5, we have t^2 = 16. Taking the square root of both sides, we get t = ±4. Since time cannot be negative in this context, the urchin will hit the rocks after 4 seconds. Therefore, the correct answer is 4s.

3)

Which table of values could be used to solve –2x² – 5x + 3 = 0?

Table A

Table B

Table C

Table D

Table A could be used to solve the equation –2x2 – 5x + 3 = 0 because it likely contains a set of x-values and their corresponding y-values that satisfy the equation when plugged into it. By substituting the x-values from Table A into the equation and calculating the corresponding y-values, one can determine if any of the y-values are equal to zero, indicating a solution to the equation.

Explanation

Table A could be used to solve the equation –2x2 – 5x + 3 = 0 because it likely contains a set of x-values and their corresponding y-values that satisfy the equation when plugged into it. By substituting the x-values from Table A into the equation and calculating the corresponding y-values, one can determine if any of the y-values are equal to zero, indicating a solution to the equation.

4) The area, in m² of a rectangular deck is modelled by the function A(x) = 84x – 2x². What dimensions will give the maximum area for the deck?

21 x 21 m

21 x 42 m

21 x 63 m

21 x 882 m

The function A(x) represents the area of the rectangular deck, where x is the length of one side of the deck. The function A(x) = 84x - 2x^2 is a quadratic function with a negative coefficient for the x^2 term. This means that the graph of the function is an upside-down parabola. The maximum area of the deck will occur at the vertex of this parabola. To find the x-coordinate of the vertex, we can use the formula x = -b/2a, where a = -2 and b = 84. Plugging in these values, we get x = -84/(2*(-2)) = 21. Therefore, the dimensions that will give the maximum area for the deck are 21 x 42 m.

Explanation

The function A(x) represents the area of the rectangular deck, where x is the length of one side of the deck. The function A(x) = 84x - 2x^2 is a quadratic function with a negative coefficient for the x^2 term. This means that the graph of the function is an upside-down parabola. The maximum area of the deck will occur at the vertex of this parabola. To find the x-coordinate of the vertex, we can use the formula x = -b/2a, where a = -2 and b = 84. Plugging in these values, we get x = -84/(2*(-2)) = 21. Therefore, the dimensions that will give the maximum area for the deck are 21 x 42 m.

5) A skateboard company models its profit with the function P(x) = –2x² + 13x – 15, where x of the number, in thousands that the company sells, and P(x) is the profit in tens of thousands of dollars. How many skateboards must the company sell to break even? Use factoring to solve

At 1500 and 5000 skateboards

At 150 and 500 skateboards

At 1.5 and 5 skateboards

At 1500 and 50 000 skateboards

The question asks for the number of skateboards the company must sell to break even. To find this, we need to set the profit function P(x) equal to zero and solve for x. By factoring the quadratic equation -2x^2 + 13x - 15 = 0, we can determine that the solutions are x = 1.5 and x = 5. However, since the company sells in thousands, we need to multiply these values by 1000. Therefore, the company must sell 1500 and 5000 skateboards to break even. The answer provided, "At 1500 and 50,000 skateboards," is incorrect as it includes an extra zero in the second value.

Explanation

The question asks for the number of skateboards the company must sell to break even. To find this, we need to set the profit function P(x) equal to zero and solve for x. By factoring the quadratic equation -2x^2 + 13x - 15 = 0, we can determine that the solutions are x = 1.5 and x = 5. However, since the company sells in thousands, we need to multiply these values by 1000. Therefore, the company must sell 1500 and 5000 skateboards to break even. The answer provided, "At 1500 and 50,000 skateboards," is incorrect as it includes an extra zero in the second value.

6)

Use the graph of the function below to solve the corresponding quadratic equation.

X = –16 and x = 0

X = 0.8 and x = –4

X = 0.8 and x = 0

X = 4 and x = 1.25

The correct answer is x = 0.8 and x = –4. This means that the quadratic equation represented by the graph has two solutions, one at x = 0.8 and the other at x = –4. These values of x correspond to the points where the graph intersects the x-axis.

Explanation

The correct answer is x = 0.8 and x = –4. This means that the quadratic equation represented by the graph has two solutions, one at x = 0.8 and the other at x = –4. These values of x correspond to the points where the graph intersects the x-axis.

7) Use factoring to solve 3x² – 13x – 10 = 0

X = 3/2 and x = 5

X = 2/3 and x = –5

X = –2/3 and x = –5

X = –2/3 and x = 5

To solve the quadratic equation 3x^2 - 13x - 10 = 0 using factoring, we need to find two numbers that multiply to give -30 (the product of the coefficient of x^2 and the constant term) and add up to -13 (the coefficient of x). The numbers that satisfy these conditions are -15 and 2. Therefore, we can rewrite the equation as (3x + 2)(x - 5) = 0. Setting each factor equal to zero, we get 3x + 2 = 0 and x - 5 = 0. Solving these equations gives x = -2/3 and x = 5 as the solutions.

Explanation

To solve the quadratic equation 3x^2 - 13x - 10 = 0 using factoring, we need to find two numbers that multiply to give -30 (the product of the coefficient of x^2 and the constant term) and add up to -13 (the coefficient of x). The numbers that satisfy these conditions are -15 and 2. Therefore, we can rewrite the equation as (3x + 2)(x - 5) = 0. Setting each factor equal to zero, we get 3x + 2 = 0 and x - 5 = 0. Solving these equations gives x = -2/3 and x = 5 as the solutions.

8)

Which graph could you use to solve –2x² – 10x – 5 = 7

Graph A

Graph B

Graph C

Graph D

Graph C could be used to solve the equation –2x^2 – 10x – 5 = 7 because it appears to be a quadratic graph. The equation is in the form of a quadratic equation, and graph C shows a parabolic shape, which is characteristic of quadratic functions. By analyzing the graph, one can determine the x-intercepts or solutions of the equation.

Explanation

Graph C could be used to solve the equation –2x^2 – 10x – 5 = 7 because it appears to be a quadratic graph. The equation is in the form of a quadratic equation, and graph C shows a parabolic shape, which is characteristic of quadratic functions. By analyzing the graph, one can determine the x-intercepts or solutions of the equation.

9) The population of a city, P(t), is modelled by the quadratic function P(t) = 8t² + 680t + 13200, where t is the time in years. Note t = o corresponds to the year 2000. According to the model, in what year will the population be 30 000?

2100

2021

2121

2210

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Explanation

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