NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Quadratic graphs and equations

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A sports coach is allocating training equipment across 75 storage bins arranged in a row. The number of items in bin x is given by the quadratic function y = x^2 - 44x + 400.

At which bin number does the minimum number of items occur? Give your answer as an exact value.

Concept illustrations

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A parabola with two real roots. The graph y = x² − 9 crosses the x-axis at −3 and 3. These are the solutions of x² − 9 = 0; the vertex is (0,−9).xy-33y = x² − 9(0,−9)
The graph y = x² − 9 crosses the x-axis at −3 and 3. These are the solutions of x² − 9 = 0; the vertex is (0,−9).
An algebraic area model for two brackets. The four areas in (x + 2)(x + 3) are x², 3x, 2x and 6. Adding them gives x² + 5x + 6. Schematic algebraic lengths.x3x2x²3x2x6(x + 2)(x + 3)
The four areas in (x + 2)(x + 3) are x², 3x, 2x and 6. Adding them gives x² + 5x + 6. Schematic algebraic lengths.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The quadratic function is given as y = x^2 - 44x + 400, where y is the number of items in bin x.
  2. For a quadratic function in the form y = ax^2 + bx + c, the vertex (which gives the minimum or maximum) occurs at x = -b/(2a).
  3. Here, a = 1 and b = -44, so substitute into the vertex formula: x = -(-44)/(2 * 1).
  4. Simplify the expression: x = 44 / 2.
  5. Calculate: 44 ÷ 2 = 22.
  6. Since a = 1 > 0, the parabola opens upwards, so the vertex at x = 22 gives the minimum value.
  7. The bin number where the minimum occurs is x = 22.
  8. Verify by checking values near x = 22: at x = 21, y = 21^2 - 44*21 + 400 = 441 - 924 + 400 = -83; at x = 22, y = 484 - 968 + 400 = -84; at x = 23, y = 529 - 1012 + 400 = -83.
  9. The value at x = 22 is lower than at x = 21 and x = 23, confirming it is the minimum.
  10. The minimum occurs at bin number 22.
  11. The result is an exact integer, so no rounding is needed.
  12. Therefore, the minimum number of items occurs at bin number 22.

Reference answer: 22

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