Help with problem solving

**greenbean** · March 28th 2008, 09:58 PM

Please help me with the following question:

A sculptor commissioned to design a monument for the long reach city council, has chosen a parabolic shape that stand 15m high with supports at 45° as shown in the diagram. For aesthetic reason the sculptor chooses the shape given by the function

Y = 15 - x²
π (pie)

ie. y equals 15 minus x² over π (pie)

Find the length of the support beam marked A. Provide justification for your answer.

**TheEmptySet** · March 28th 2008, 10:02 PM

do you mean

$E_1 \rightarrow y=\frac{15-x^2}{\pi}$

or

$E_2 \rightarrow y=15-\frac{x^2}{\pi}$

**TheEmptySet** · March 28th 2008, 10:16 PM

$\tan^{-1}(45^{\circ})=1$

so the equation of the line is $y=x$

setting $y=x$ equal to $y=15-\frac{x^2}{\pi}$

$x=15-\frac{x^2}{\pi} \iff x^2+\pi x=15 \pi$

so I will solve by completeing the square so we will add $\frac{\pi^2}{4}$ to both sides

$x^2+\pi x +\frac{\pi^2}{4}=15 \pi +\frac{\pi^2}{4} \iff (x-\frac{\pi}{2})^2=\frac{\pi(60+\pi)}{4}$

$x=\frac{\pi}{2} +\frac{\sqrt{\pi(60+\pi)}}{2}$

So this is half the distance from the diagram so if we multiply it by 2 we will get the length of the line.

$L=\pi+\sqrt{\pi(60+\pi)}$

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