Differentiation of transcendental functions?

ABCD is a trapezium with AB = CD, with vertices on the circle and with centre O. AD is a diameter of the circle. The radius of the circle is 4 units.

a. Find BC in term of theta.

b. Find the area of the trapezium in terms of theta and hence find the maximum area.

Image of the diagram for this question:

http://i45.tinypic.com/23s944h.png

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Re: Differentiation of transcendental functions?

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**tutto00** ABCD is a trapezium with AB = CD, with vertices on the circle and with centre O. AD is a diameter of the circle. The radius of the circle is 4 units.

a. Find BC in term of theta.

b. Find the area of the trapezium in terms of theta and hence find the maximum area.

Image of the diagram for this question:

http://i45.tinypic.com/23s944h.png

1. The area of a trapezium is calculated by: $\displaystyle area = \frac{top + base}2 \cdot height$

2. Replace:

$\displaystyle base = 2r$

$\displaystyle height = r \cdot \sin(\theta)$

$\displaystyle top = 2 r \cdot \cos(\theta)$

3. The area a is a function of $\displaystyle \theta$ while r is a constant:

$\displaystyle a(\theta) = r^2 \sin(\theta) (1+\cos(\theta))$

4. Differentiate wrt $\displaystyle \theta$ and solve $\displaystyle a'(\theta) = 0$ for $\displaystyle \theta$