How to solve situational problems involving parabola? (please help!)

Question

A gateway is a parabolic shape whose vertex is 20 𝑓𝑡. from the ground, (0,20), and the width on the ground is 25 𝑓𝑡. What is the maximum height of a vehicle to cross the gateway if it is 7 𝑓𝑡 wide? Find the standard equation the models the gateway.

A TIP OR A HINT WILL REALLY HELP ME! thank you.

Answer 1

graphically the downward opening parabola is y=a(x-0)^2 +20 in vertex form with a<0

2 points on the parabola are (12.5,0) and (-12.5,0), axis of symmetry: x=0

plug in either point to solve for a

0=a(12.5)^2+20

a =-20/12.5^2

y =-(20/156.25)x^2 +20

two more points on the parabola are (7/2,y) and (-7/2,y)

where y is the maximum height of a truck that's 7 feet wide, that can get through the parabola shaped tunnel

y=-(20/156.25)3.5^2 + 20 = -20(12.25)/156.25 + 20 = -245/156.25+20 = 20-1.568 = 18.432 feet high maximum

(that assumes no oncoming traffic, as the truck hogs both lanes & any shoulders)

with luck there're no arithmetic mistakes above, but this is the general solution method