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an engineer is designing a parabolic arch. the arch must be 15 m high, and 6 m wide at the height of 8 m. Determine a quadratic function that satisfies these conditions.  

Verified Answer :

Let's assume we've got our maximum at 0 (this simplifies things). We need f(0)=15.
Also, to make it 6m wide at a height, we have to set f(3)=8 and f(-3)=8. Now we have:
\(c=15 \\ 9a+3b+c=8 \\ 9a-3b+c=8\)
We could easily notice that b=0 and c=15, let's find a: 9a+15=8 or 9a=-7 or \(a=-\frac{7}{9}\)
The function is \(f(x)=-\frac{7}9x^2+15\)

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