SOLUTION: A truck is about to pass through a one-way tunnel in the form of a semi-ellipse, which 15 m across and 4 m high in the middle. If the truck has a width of 4 m and a height of 3.5 m
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-> SOLUTION: A truck is about to pass through a one-way tunnel in the form of a semi-ellipse, which 15 m across and 4 m high in the middle. If the truck has a width of 4 m and a height of 3.5 m
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Question 1171631: A truck is about to pass through a one-way tunnel in the form of a semi-ellipse, which 15 m across and 4 m high in the middle. If the truck has a width of 4 m and a height of 3.5 mwill be able to pass through the tunnel ? Justify your answer . Answer by htmentor(1343) (Show Source):
You can put this solution on YOUR website! Standard form for an ellipse centered at the origin is:
(x/a)^2 + (y/b)^2 = 1, where a, b are the semi-major and semi-minor
axes, respectively. Since the width of the tunnel = 15, a = 7.5.
The height = b = 4.
Thus the equation describing the tunnel opening is:
(x/7.5)^2 + (y/4)^2 = 1
A truck having width 4 and height 3.5 will be able to pass through
the tunnel. We can see this mathematically and graphically.
Mathematically, we only need to consider if the sides of the
truck clear the topof the tunnel, since that would be the
lowest point. The sides of the truck are at x = +-2.
We need the height of the tunnel at these points:
(y/4)^2 = 1 - (2/7.5)^2 -> y = 4*sqrt(1-(2/7.5)^2) = 3.855 m.
Since this is greater than the truck height, the truck can pass through.
We can also see this graphically:
