Porisma

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Optimise a rectangle inside a semicircle

The diagram shows rectangle PQRSPQRS inscribed in a semicircle of radius 1010 cm, with P,QP,Q on its diameter and S,RS,R on its arc. Given PQ=xPQ=x cm, find the perimeter pp cm in terms of xx.

Show that its area is A=x2400−x2 cm2A=\dfrac{x}{2}\sqrt{400-x^2}\text{ cm}^2. Find the exact value of pp when AA has its stationary value.

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Rectangle PQRS inside a semicircle of radius 10 cm. P and Q lie on the horizontal diameter; S and R are the upper left and upper right arc points. PQ has length x. A radius from the centre to S and a half-width x/2 form the right triangle used to find the height.
Rectangle PQRS inside a semicircle of radius 10 cm. P and Q lie on the horizontal diameter; S and R are the upper left and upper right arc points. PQ has length x. A radius from the centre to S and a half-width x/2 form the right triangle used to find the height.
Rectangle PQRS inside a semicircle of radius 10 cm. P and Q lie on the horizontal diameter; S and R are the upper left and upper right arc points. PQ has length x. A radius from the centre to S and a half-width x/2 form the right triangle used to find the height.

[2008/JJC/Promo/12]

CJC · Discussion · Question 11

Use half the width in the radius triangle, then differentiate the area rather than the perimeter.

Read the worked solution