Question

Bill is on his first day at work delivering furniture for a large store in Calgary. He is unfamiliar with the city and in route to his first delivery, he encounters an archway that supports an overpass above him. The maximum height of the tunnel is 8 meters and follows a parabolic curve: (y = -x² + 4x). Bill decides to go through the tunnel knowing that his truck is 2.5m wide and only 5m tall. There are two lanes through the tunnel, but traffic is heavy, and he is forced to stay in his own lane. Will Bill make it through the tunnel, or will this be his last day on the job? Explain.

a) Bill will make it through the tunnel because his truck is shorter than the maximum height.
b) Bill will not make it through the tunnel because his truck is too wide.
c) Bill will make it through the tunnel because his truck is narrow enough.
d) Bill will not make it through the tunnel because his truck is taller than the maximum height.

Answer 1

Bill's truck will not make it through the tunnel as it is taller than the maximum height of the tunnel at the point where the truck would travel within its lane.

Step-by-step explanation:

To determine whether Bill's truck will make it through the tunnel, we need to examine the height of the tunnel at the points where the truck will be when it is passing through. The truck is 2.5 meters wide, so we need to calculate the height of the tunnel 1.25 meters from the centerline (since traffic requires him to stay in his lane and he cannot be perfectly centered). The parabolic equation of the tunnel is given as y = -x² + 4x. We substitute x = 1.25 into the equation:

• y(1.25) = -(1.25)² + 4(1.25)
• y(1.25) = -1.5625 + 5
• y(1.25) = 3.4375 meters

This means that at 1.25 meters from the center, the tunnel is 3.4375 meters tall. Since Bill's truck is 5 meters tall, it exceeds the height of the tunnel at the point where it would be traveling, and thus, Bill should not attempt to pass through this tunnel.