I have the quadratic function
$$y=\frac{1}{294}(x-84)^2-24$$
that represents the shape of the cables of a certain bridge. I am suppose to be determining the vertical height of the cables above the minimum at a point that is $35$m horizontally from one of the towers.
I keep getting an answer of $-15.83$m, when I should be getting an answer of $8.17$m. Any suggestions or tips would be much appreciated.
On
The minimum of the function, $$f(x)=\frac{1}{294}(x-84)^2-24,$$ is reached when $f'(x)=0$, so $$f'(x)=\frac{1}{147}(x-84) = 0 \iff x = 84,$$ and the value of $f$ at this point equals $-24$. Another way to see it is noting that $(x-84)^2\geq 0$ for every $x$ and so $f(x)\geq -24$ for every $x$. Then the height to the ground depends on where are the towers. If the the tower $T$ is at a distance $d$ from the lowest point and of height $h$ you get that the minimum point is at $$h_0 = h-(f(84 \pm d)-f(84))$$ (meter) from the ground.
The minimum point is at $-24m$, the height you have calculated is $-15.83m$. This is $-15.83-(-24)=8.17m$ above the minimum point.