SOLUTION: A semielliptic archway has a height of 20feet and a width of 50feet, can a truck 14feet high and 10feet wide drive under the arch without going into the other lane?

Question 481812: A semielliptic archway has a height of 20feet and a width of 50feet, can a truck 14feet high and 10feet wide drive under the arch without going into the other lane?
Answer by Edwin McCravy(20060) (Show Source): You can put this solution on YOUR website!
We'll put the origin at the center of the roadway.
The major axis is 50. so the semi-major axis = a = 25
The semi-minor axis is 20.
So the equation of the ellipse is
or or
The graph of the whole ellipse is:

Here is the semielliptic archway:

If we put the truck so that its left side is right on the center line
the truck will fit in this rectangle:

We will see how much clearence it has if its left side is right on the
center line:
We substitute x = 10 in

Multiply thru by 400

Since the truck is 14 ft. high, it will have a clearance of
about 4.33 feet if its left side is right on the center line.
Actually we don't want the truck to be right on the center
line, so let's see how far to the right the truck can be
before its upper right side would scrape the archway.
We sustitute 14 for y in the ellipse equation:

Multiply through by 625

So the truck could easily pass under the archway without going into
the left lane, as lang as the right side is less than 17.85 feet
right of the center line, which means the left side must be closer
than 7.85 feet right to the center line, which will be very easy
for the driver to do.

Edwin

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