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Quadratic Real-World Applications

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Quadratic Real-World Applications

Solve each problem using the information given. Show your working for calculation questions and include units where appropriate.

Build and interpret quadratic models

1.For the sports-day long jump, Grace models the height of a practice ball with h = -2t² + 8t + 1, where h is in metres and t is seconds after the throw. Which statement best describes the vertex of this parabola?

The ball starts 8 metres high.
The ball reaches its maximum height of 9 metres after 2 seconds.
The ball hits the ground after 2 seconds.
The ball travels 9 metres horizontally.

2.At a sports-day stall, Omar sets up a rectangular banner area against a wall, so only three sides need rope. There are 24 metres of rope. If each short side is w metres, the long side is (24 - 2w) metres. Write a quadratic rule for the area A in terms of w, then state the practical domain for w.

3.Mei wants the banner area from Question 2 to be as large as possible. Find the value of w that gives the maximum area and calculate that area.

4.A ball thrown during the athletics demonstration has height h = -5t² + 15t + 2 metres after t seconds. Calculate the time at which it reaches its maximum height and the maximum height.

Use roots, intercepts and context

5.A foam javelin’s height above the ground is modelled by h = -5t² + 20t + 25. Factorise the expression and find when the javelin hits the ground. Explain why one solution is not physically relevant.

6.A ticket stall models its sports-day profit by P = -2x² + 80x - 600, where x is the number of ticket bundles sold and P is profit in dollars. Find the two break-even sales levels by solving P = 0.

7.For the profit model P = -2x² + 80x - 600, calculate the number of bundles that gives maximum profit and find that profit. Interpret your result in context.

8.True or false? If a quadratic model has a negative coefficient for x², its graph opens downwards and its vertex is a maximum point. Give a brief reason.

TrueFalse

9.Grace says, “The two roots of a height-time quadratic always both describe times when the object hits the ground.” Is her statement always correct? Use h = -5t² + 20t + 25 to support your answer.

10.Create a sports-day situation that could be modelled by a quadratic function. Write a possible quadratic rule, identify what its variables represent, and describe what the vertex would mean in your situation.