Question

A bridge underpass in the shape of an elliptical arch, that is, half of an ellipse, is 40 feet wide and 13 feet high. An eight foot wide rectangular truck is to drive (safely) underneath. How high can it be?

Answer 1

the truck should not be higher than 12.934 ft

Explanation:

if the bridge underpass is half of an ellipse , then the arch function will be B(x,y) such that

x²/a² + y²/b² = 1 , for x≥0 and y≥0 and y=0 for x≤0

where x= width , y= height

for

a= length of the bridge underpass = 40 ft

b= height of the bridge underpass = 13 ft

x²/(40 ft)² + y²/(13 ft)² = 1

that represents all the points of the arch

the maximum height is reached when the truck touches the arch, then

for a point x= 4 ft wide , since 8 foot wide represents 4 foot to the left (x=-4) and 4 foot to the right (x=4)

then

(4 ft)²/(40 ft)² + y²/(13 ft)² = 1

1/100 - y²/(13 ft)² = 1

y/13 ft = √(1-1/100) = 3/10*√11

y= 3/10*√11 * 13 ft = 12.934 ft

y= 12.934 ft

then the truck should not be higher than 12.934 ft