2022 THREE(e)

Excellence
Question
Megan cycles from her home, H, to school, S, each day.
Loading diagram...
She rides along a path from her home to point P at a constant speed of 10 kilometres per hour.
At point P, Megan cuts across a park, heading directly to school. When cycling across the park,
Megan can only cycle at 6 kilometres per hour.

At what distance from her home should she choose to cut across the park in order to make her
travelling time a minimum?

You must use calculus and show any derivatives that you need to find when solving this problem.
Official Answer
Total time == time (HP) ++ time (PS)
Method A
Let x=x= distance PQ

T=4−x10+x2+46T=\frac{4-x}{10}+\frac{\sqrt{x^2+4}}{6}

dTdx=−110+12(x2+4)−12⋅2x6\frac{dT}{dx}=-\frac{1}{10}+\frac{\frac{1}{2}(x^2+4)^{-\frac{1}{2}}\cdot 2x}{6}

dTdx=−110+x6x2+4\frac{dT}{dx}=-\frac{1}{10}+\frac{x}{6\sqrt{x^2+4}}

For maximum/minimum time, dTdx=0\frac{dT}{dx}=0

110=x6x2+4\frac{1}{10}=\frac{x}{6\sqrt{x^2+4}}

6x2+4=10x6\sqrt{x^2+4}=10x

x2+4=106x\sqrt{x^2+4}=\frac{10}{6}x

x2+4=259x2x^2+4=\frac{25}{9}x^2

4=169x24=\frac{16}{9}x^2

3616=x2\frac{36}{16}=x^2

x=1.5x=1.5

4−1.5=2.54-1.5=2.5

Megan should travel 2.5 km along the path
before cutting across the park.
Method B
Let x=x= distance HP

T=x10+((4−x)2+4)6T=\frac{x}{10}+\frac{\sqrt{((4-x)^2+4)}}{6}

dTdx=110+x−46x2−8x+20\frac{dT}{dx}=\frac{1}{10}+\frac{x-4}{6\sqrt{x^2-8x+20}}

dTdx=0\frac{dT}{dx}=0

110+x−46x2−8x+20=0\frac{1}{10}+\frac{x-4}{6\sqrt{x^2-8x+20}}=0

5(x−4)=−3x2−8x+205(x-4)=-3\sqrt{x^2-8x+20}

25(x2−8x+16)=9(x2−8x+20)25(x^2-8x+16)=9(x^2-8x+20)

25x2−200x+400=9x2−72x+18025x^2-200x+400=9x^2-72x+180

16x2−128x+220=016x^2-128x+220=0

x=2.5x=2.5 or 5.55.5

Since x<4x<4, x=2.5x=2.5 km
Grading Criteria

Achievement (u)

-

Merit (r)

  • Correct dTdx\frac{dT}{dx}.

Excellence T1

  • T1 Method A
    x=1.5x=1.5 found with correct derivative

    OR

    T1 Method B
    x=2.5x=2.5 or 5.55.5 found (5.55.5 not discarded) with correct derivative.

Excellence T2

  • T2 Correct solution with correct derivative.
Video Explanation
2022 NCEA L3 Calculus Exam Walkthrough by infinityplusone(starts at 69:00)
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Exam Paper (2022)Assessment Schedule (2022)
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