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

Name Date

Where two graphs meet Problem set B

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

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. The curve \(y = 3x^2 − 2x − 4\) and the line \(y = 2x + 5\) meet at two points. Find the coordinates of both points.

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

  3. A model is \(g(x) = 4 + 15 × 0.7^x\).

    1. (a) Find the \(y\)-intercept of the graph of \(g\).
    2. (b) Write down the equation of the horizontal asymptote.
    3. (c) Solve \(g(x) = 6\).

In context

  1. A tennis ball is hit upwards from a balcony. Its height above the ground after \(t\) seconds is \(h = −4.9t^2 + 9t + 6.2\) metres.

    1. (a) Find the greatest height and the time at which it is reached.
    2. (b) Find when it hits the ground.
    3. (c) For how long is it higher than 9 m?

  2. A bakery sells \(x\) cakes a week. The weekly cost is \(C = 950 + 8x\) and the weekly revenue is \(R = 46x − 0.2x^2\), both in dollars.

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

  3. The value of a laptop is \(V = 1800 × 0.78^t\) euros after \(t\) years. A student's savings are \(S = 300 + 420t\) euros, for \(0 \leq t \leq 6\).

    1. (a) Find when the two values are equal.
    2. (b) Which is greater after 4 years? Give both values.

  4. A workshop makes \(n\) candles in one batch. The average cost of one candle, in euros, is \(A = 180/n + 0.5n + 4\).

    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 \(y = x^2 + 2x − 3\) and the line \(y = −x + 4\) meet at points A and B.

    −6−4−22−55101520xyAB
    1. (a) Find the coordinates of A and B.
    2. (b) Find the length of AB.

  2. A rocket is launched from the ground. Its height is \(h_1 = −4.9t^2 + 25t\) metres after \(t\) seconds. At the same moment a balloon rises steadily from a height of 5 m, so its height is \(h_2 = 5 + 3t\) metres.

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

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