Question 751408
The parabola is upside down and has the vertex at (0,10). Notice how this is the highest point.


The two roots are (-9,0) and (9,0). The horizontal distance between these two points is exactly 18 m. Basically how I found these two points is that I divided the distance of 18 m in half to get 9 m, then you walk 9 m in each direction to land on (-9,0) and (9,0)


So we have the three points: (0,10), (-9,0) and (9,0)


The first point is the vertex. The first point is also the y-intercept.


The next two points are the x-intercepts or roots.


Let's use the roots to find the equation


x = -9 or x = 9


x+9 = 0 or x-9 = 0


k(x+9)(x-9) = 0 ... for some fixed number k (we don't know it yet, but we'll find it soon)


k(x^2 - 81) = 0


The equation so far is y = k(x^2 - 81)


Since the y-intercept is (0,10), we can plug in x = 0 and y = 10 to find k


y = k(x^2 - 81)


10 = k(0^2 - 81)


10 = k(0 - 81)


10 = k(-81)


10 = -81k


-81k = 10 


k = 10/(-81)


k = -10/81


So the equation that models the tunnel is {{{y = -expr(10/81)(x^2-81)}}} which distributes to {{{y = -expr(10/81)(x^2)-expr(10/81)(-81)}}} and that simplifies to {{{y = -expr(10/81)(x^2)+10}}}


So the equation in standard form is {{{y = -expr(10/81)x^2+10}}}


Now we're told that the truck is 7.5 m, so this means that the height y is 7.5, so we can plug in y = 7.5 and solve for x to find the clearance



{{{y = -expr(10/81)x^2+10}}}


{{{7.5 = -expr(10/81)x^2+10}}}


{{{7.5-10 = -expr(10/81)x^2}}}


{{{-2.5 = -expr(10/81)x^2}}}


{{{-2.5*81 = -10x^2}}}


{{{-202.5 = -10x^2}}}


{{{-202.5/(-10) = x^2}}}


{{{20.25 = x^2}}}


{{{x^2 = 20.25}}}


{{{x = sqrt(20.25)}}} or {{{x = -sqrt(20.25)}}}


{{{x = 4.5}}} or {{{x = -4.5}}}


So when {{{x = 4.5}}} or {{{x = -4.5}}}, the height is exactly 7.5 m, which means that the truck's width must be less than 2*4.5 = <font color="red">9 feet</font> across so it can fit in the tunnel.