Question

An arch in the shape of a parabola has a height of 50 feet at the center and a span of 120 feet across. Find the height of the arch 25 feet from the center.

(Solve the problem. Type only a number for your answer. 80 points!!!

Answer 1

Check the picture below.

so, we know the parabola has a vertex at (0 , 50), we also know it passes through (60 , 0), so then

`~~~~~~\textit{vertical parabola vertex form} \\\\ y=a(x- h)^2+ k\qquad \begin{cases} \stackrel{vertex}{(h,k)}\\\\ \stackrel{`

`\stackrel{vertex}{(0~~,~~50)}\implies y=a(x-0)^2+50\implies y = ax^2+50 \\\\\\ \begin{cases} x =60\\ y = 0 \end{cases}\implies 0=a6^2+50\implies -50=3600a\implies -\cfrac{50}{3600}=a \\\\\\ -\cfrac{1}{72}=a~\hfill therefore~\hfill \boxed{y=-\cfrac{1}{72}x^2+50}`

so then, what's "y" when x = 25?

`y=-\cfrac{1}{72}25^2+50\implies y = -\cfrac{625}{72}+50\implies y = \cfrac{2975}{72}\implies y\approx 41.32`