A tunnel has height, h, (in metres) given by h=14-x^2 where x is the horizontal distance from the centre of the tunnel. Find the cross sectional area of the tunnel. Also find the maximum height of a truck passing through the tunnel that is 4m wide.
Firstly, solve 0=14-x^2 to find the horisontal distance to the edges of the tunnel. x1=sqrt(14), x2= -sqrt(14).
Integrate h=14-x^2 between x1 and x2 28*sqrt(14) -(2(sqrt(14)^3))/3. This is the required area
Next, the center of the tunnel is the heighest point so we would place the center of the truck here. Threfore, the edges of the truck are at x=2 and x=-2. The height of the tunnel here is 14-(2^2) = 14-((-2)^2) = 14-4 = 10. Therefore 10 is the max height.
JG
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