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

Question 481494: A semielliptic archway has a height of 20 feet and a width of 50 feet, can a truck 14 feet high and 10 feet wide drive under the archway without going into the other lane?
Answer by lwsshak3(11628) (Show Source): You can put this solution on YOUR website!
A semielliptic archway has a height of 20 feet and a width of 50 feet, can a truck 14 feet high and 10 feet wide drive under the archway without going into the other lane?
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Draw an ellipse with a horizontal major axis of the standard form:
(x-h)^2/a^2+(y-k)^2/b^2=1
Place the center of this ellipse at the origin (0,0).
The equation now becomes: x^2/a^2+y^2/b^2=1
The given width of the semielliptic archway of 50 feet represents the major axis=2a
a=25
a^2=625
The given height of the semielliptic archway of 20 feet represents half of the major axis=b
b=20
b^2=400
..
Equation of the ellipse:
x^2/625+y^2/400=1
The top half of this ellipse represents the semielliptic archway.
,..
Determine whether a truck 14 feet high and 10 feet wide can drive under the archway without going into the other lane?
..
Plug y=14 ft (height of truck) into the equation and solve for x (allowable width of archway).
x^2/625+14^2/400=1
x^2/625+196/400=1
x^2=(1-196/400)*625=318.75
Taking the positive sqrt because using upper half of ellipse
x=√318.75=17.85 ft
This means that at the height clearance of 14 ft, there is more than 17 ft of road width allowed for the 10 ft wide truck.
ans:
So, the archway can easily accommodate the 14 ft high and 10 ft wide truck.
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