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Further applications of differentiation — Question 338

Original QCE Vault practice · 10 marks

Q338 · Practice questionTechnology-activeComplex unfamiliar10 marks

QUESTION 338 (10 marks)

A festival entrance has upper profile y=4sin⁡(πx/12)y=4\sin(\pi x/12) metres for 0≤x≤120\leq x\leq12. A rectangular screen stands on the baseline and is centred at x=6x=6. Its top corners touch the profile, and its height is hh metres, with 0<h<40<h<4.
Arch y=4 sin(pi x/12) on [0,12]. A rectangular screen is centred at x=6, rests on the baseline and has top corners on the arch. A vertical double arrow marks height h.
a)
Show that the screen width is W(h)=12−24πarcsin⁡(h/4)W(h)=12-\dfrac{24}{\pi}\arcsin(h/4).
[2 marks]
b)
Put u=arcsin⁡(h/4)u=\arcsin(h/4). Show that a stationary screen area satisfies tan⁡u=π/2−u\tan u=\pi/2-u.
[3 marks]
c)
Determine the maximum screen area and its dimensions, each to two decimal places. Justify that this is a global maximum.
[3 marks]
d)
Determine the percentage of the entrance area covered by the largest screen, to one decimal place.
[2 marks]
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