2018 THREE(c)

Merit
Question
The diagram below shows the graph of the function y=15−x2y = 15 - x^2, inside which an isosceles
triangle OAB has been drawn.
Loading diagram...
Find the maximum possible area, AA, of the triangle.
You may assume that your answer is a maximum.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*
Official Answer
Area =12⋅2x⋅(15−x2)=15x−x3=\dfrac{1}{2}\cdot 2x\cdot (15-x^2)=15x-x^3

dAdx=15−3x2\dfrac{dA}{dx}=15-3x^2

Max when dAdx=0\dfrac{dA}{dx}=0

3(5−x2)=03(5-x^2)=0

x=±5x=\pm\sqrt{5}
y=10y=10

Area =12×25×10=\dfrac{1}{2}\times 2\sqrt{5}\times 10

=105  (=22.36)=10\sqrt{5}\;(=22.36)
Grading Criteria

Achievement (u)

  • Correct dAdx\dfrac{dA}{dx}

Merit (r)

  • Correct solution with correct derivative.

Excellence T1

-

Excellence T2

-
Video Explanation
NCEA Level 3 Calculus Differentiation 2018 NZQA Exam - Worked Answers by infinityplusone(starts at 42:40)
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Exam Paper (2018)Assessment Schedule (2018)
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