NCEA Calculus 2023 THREE(c - Solution & Explanation

Question

Char goes for a ride on a Ferris wheel. As she rotates around,
her position can be described by the pair of parametric
equations :

x = 5\sqrt{2}\,\sin\left(\frac{\pi t}{5}\right)

and

y = 10 - 5\sqrt{2}\,\cos\left(\frac{\pi t}{5}\right)

where

t

is time, in seconds, from the start of the ride.

Find the gradient of the normal to this curve at the point when

t = 6.25

seconds, after the start of the ride.

*You must use calculus and show any derivatives that you need
to find when solving this problem.*

Official Answer

\dfrac{dx}{dt}=\sqrt{2}\pi\cos\left(\dfrac{\pi t}{5}\right)

\dfrac{dy}{dt}=\sqrt{2}\pi\sin\left(\dfrac{\pi t}{5}\right)\Rightarrow

\dfrac{dy}{dx}=\tan\left(\dfrac{\pi t}{5}\right)

When

t=6.25\Rightarrow

\dfrac{dy}{dx}=\tan(1.25\pi)=1

Normal gradient:

m=-1

Grading Criteria

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Video Explanation

2023 NCEA L3 Calculus Exam Walkthrough by infinityplusone(starts at 54:28)

2023 THREE(c)