SOLUTION: 1. A one-way tunnel has the shape of a semi-ellipse that is 12 ft. high at the center and 28 ft. across. a) Will a truck that is 8 ft. across and 10 ft. high will be able to pas

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: 1. A one-way tunnel has the shape of a semi-ellipse that is 12 ft. high at the center and 28 ft. across. a) Will a truck that is 8 ft. across and 10 ft. high will be able to pas      Log On


   

Question 1126701: 1. A one-way tunnel has the shape of a semi-ellipse that is 12 ft. high at the center and 28 ft. across.
a) Will a truck that is 8 ft. across and 10 ft. high will be able to pass through the tunnel? (Show Solution)
(b) Suppose the road under the tunnel has two lanes, and the truck is allowed to pas through only one lane. Will it still be able to pass through the tunnel? (Show Solution)
Answer by greenestamps(13200) About Me  (Show Source):
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Place the center of the ellipse on the road in the center of the tunnel.

Then the 28-foot width of the tunnel is the major axis of the ellipse, so the semi-major axis is 14; and the 12-foot height of the tunnel is the semi-minor axis of the ellipse.

The standard form of the equation of the ellipse is then

x%5E2%2F14%5E2%2By%5E2%2F12%5E2+=+1
x%5E2%2F196%2By%5E2%2F144+=+1

To solve both parts of the problem, we need to now what the y value (height of the tunnel) is for particular x values (horizontal distances from the center of the tunnel). The equation solved for y in terms of x is

y+=+sqrt%28144%281-x%5E2%2F196%29%29

For the first question, the truck can pass through the center of the tunnel. Since the truck is 8 feet wide, we need to know the height of the tunnel 4 feet either side of the center. A graphing calculator (or manual calculations) show that the height of the tunnel there is about 11.5 feet, far higher than needed, making it possible for the 10-foot truck to pass through easily.

For the second question, the truck has to stay on its half of the tunnel, so we need to know the height of the tunnel 8 feet either side of the center. This height turns out to be about 9.85 feet, which means the truck will not be able to pass through the tunnel staying in its lane.