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I was wondering if someone could possibly explain this question:

"A stadium should be oblong on plan with straight sides of length h and semi-circular arcs of radius r at either end. The facade must be 10m high all the way around and as such it represents a significant part of a stadium's budget. The QS has indicated that there will only be sufficient funds for a total area of 2500m² of facade, and suggests it is therefore important to maximise the plan area of the stadium to make best use of the land available.

a) Derive an equation for the total L of the stadium facade on plan in terms of h and r

b) Derive an equation for the total plan area, A, enclosed by the stadium facade in terms of h and r

c) Hence find the values of both h and r that maximise the plan area of the stadium, and the corresponding maximum area, A, that can be enclosed without requiring more than 2500m² of facade, and describe the resulting shape."

I have done the first two parts:

a) L = 2πr + 2h

b) A = πr² + 2rh

However I am struggling with part C. I am assuming it involves differentiating A, however this gives me two unknowns of r and h and I don't know how to get rid of one of them.

GFauxPas
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S Hayward
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  • Are you familiar with the gradient operator? Lagrange multipliers? – GFauxPas May 27 '15 at 21:09
  • Hi, no I'm not. This should just be simple calculus, but I can't seem to get it right – S Hayward May 27 '15 at 21:11
  • Use the 2500 and 10 to derive an expression for. Using a) you get a first equation. Express h as function of r. Place in b) and get the value of r that maximize A. Now calculate h and A. – Moti May 28 '15 at 02:13

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