SOLUTION: A truck with a height of 190 cm enters a tunnel with a parabolic ceiling. The width of the tunnel is 20m and the maximum height of this tunnel is 10 m. At what minimal distance fro

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Question 1127352: A truck with a height of 190 cm enters a tunnel with a parabolic ceiling. The width of the tunnel is 20m and the maximum height of this tunnel is 10 m. At what minimal distance from the edge at ground level can this truck pass through the tunnel?
Answer by greenestamps(13367) About Me  (Show Source):
You can put this solution on YOUR website!


Make a graph of the tunnel with the origin at road level in the middle of the tunnel. That makes the vertex the top of the tunnel, at (0,10); with the width of the tunnel being 20m, the edges of the tunnel at road level are at (-10,0) and (10,0).

The equation of a parabola with vertex at (0,10) is

y+=+ax%5E2%2B10

To find the value of the constant a, use either edge of the tunnel at road level.

0+=+a%2810%5E2%29%2B10
0+=+100a%2B10
100a+=+-10
a+=+-1%2F10

The equation of the parabola is

y+=+%28-1%2F10%29x%5E2%2B10

We want to know the value of x when the y value (the height of the truck) is 190cm = 1.9m.

1.9+=+%28-1%2F10%29x%5E2%2B10
%281%2F10%29x%5E2+=+10-1.9+=+8.1
x%5E2+=+81
x+=+9

The truck can pass through the tunnel 9m from the center of the tunnel.

The problem asks for the minimum distance from the edge of the tunnel that the truck can pass through; 9m from the center is 1m from the edge.

ANSWER: The truck can pass through the tunnel 1m from the edge

A graph.... Green line is the 1.9m height of the truck, showing that it can fit through the tunnel 9m from the center of the tunnel, which is 1m from the edge of the tunnel.

graph%28400%2C200%2C-12%2C12%2C-2%2C12%2C%28-1%2F10%29x%5E2%2B10%2C1.9%29