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IB Mathematics: Applications and Interpretation HL · 12.4: Solving Equations with Technology · 12.4.6

Name Date

Where two graphs meet Problem set A

Use the calculator. Give answers to 3 significant figures unless the question says otherwise.

Difficulty: first skills applying them stretch

Skills

  1. The curve \(y = x^2 − 5x + 3\) and the line \(y = 2x − 6\) meet at two points. Find the coordinates of both points.

  2. Use your calculator to solve \(2^x = x^2\). Give every solution to 3 significant figures.

  3. A model is \(g(x) = 5 + 12 × 0.8^x\).

    1. (a) Find the \(y\)-intercept of the graph of \(g\).
    2. (b) Write down the equation of the horizontal asymptote.
    3. (c) Solve \(g(x) = 7\).

In context

  1. A stone is thrown upwards from a bridge. Its height above the river after \(t\) seconds is \(h = −4.9t^2 + 12t + 9.5\) metres.

    1. (a) Find the greatest height and the time at which it is reached.
    2. (b) Find when it hits the river.
    3. (c) For how long is it higher than 12 m?

  2. A workshop sells \(x\) toys a week. The weekly cost is \(C = 1500 + 12x\) and the weekly revenue is \(R = 60x − 0.25x^2\), both in dollars.

    1. (a) Find the break-even quantities.
    2. (b) Write down the smallest and the largest whole number of toys that give a profit.
    3. (c) Find the greatest weekly profit, using a whole number of toys.

  3. The value of a car is \(V = 24 000 × 0.85^t\) dollars after \(t\) years. The amount still owed on its loan is \(L = 21 000 − 1500t\) dollars, for \(0 \leq t \leq 8\).

    1. (a) Find when the two values are equal.
    2. (b) Which is greater after 5 years? Give both values.

  4. A print shop makes \(n\) posters in one batch. The average cost of one poster, in dollars, is \(A = 250/n + 0.8n + 3\).

    1. (a) Use your calculator to find the value of \(n\) that gives the lowest average cost, to 3 significant figures.
    2. (b) The number of items must be a whole number. Find the best whole number and the average cost for it.

Stretch

  1. The curve \(y = 0.5x^2 − 3x + 2\) and the line \(y = x − 3\) meet at points A and B.

    1234567−4−2246810xyAB
    1. (a) Find the coordinates of A and B.
    2. (b) Find the length of AB.

  2. A ball is thrown upwards from the ground. Its height is \(h_1 = −4.9t^2 + 22t\) metres after \(t\) seconds. At the same moment a drone rises steadily from a height of 3 m, so its height is \(h_2 = 3 + 4t\) metres.

    1. (a) Find the times at which the two heights are equal.
    2. (b) For how long is the ball higher than the drone?

jmaths.xyz · AI HL · 12.4.6 Where two graphs meet