SOLUTION: A tunnel through a mountain takes the shape of a semi-ellipse. The tunnel is 100 feet wide and has a maximum height at its center of 30 feet. If the clearance height for vehicles

Question 394062: A tunnel through a mountain takes the shape of a semi-ellipse. The tunnel is 100 feet wide and has a maximum height at its center of 30 feet. If the clearance height for vehicles needs to be a minimum of 16 feet, how much total width do they have available?
Answer by lwsshak3(11628) (Show Source): You can put this solution on YOUR website!
A tunnel through a mountain takes the shape of a semi-ellipse. The tunnel is 100 feet wide and has a maximum height at its center of 30 feet. If the clearance height for vehicles needs to be a minimum of 16 feet, how much total width do they have available?
Standard form of an ellipse:
x^2/a^2+y^2/b^2=1
assume the center of the ellipse is at (0,0)
a=50 (major axis)
b=30 (minor axis)
x^2/50^2+y^2/30^2=1
x^2/2500+y^2/900=1
900x^2+2500y^2=900*2500
900x^2=900*2500-2500y^2
x^2=2500-(2500/900)y^2
x^2=2500-(25/9)y^2
x=sqrt(2500-(25/9)y^2)
given: y=16 feet of minimum clearance required
x=sqrt(2500-(25/9)*16^2
=+ or- sqrt(1788.89)=+-42.30 feet
ans: total width required = 2*42.30 =84.6 feet (at this width the height of the tunnel is 16 feet)
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