The base of a certain solid is an equilateral triangle with altitude 5. Cross-se

Question

The base of a certain solid is an equilateral triangle with altitude 5. Cross-sections perpendicular to the altitude are semicircles. Find the volume of the solid, using the formula applied to the picture shown above (click for a better view), with the left vertex of the triangle at the origin and the given altitude along the x-axis. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. The lower limit of integration is a = The upper limit of integration is b = The diameter 2r of the semicircular cross-section is the following function of x: A(x)= Thus the volume of the solid is V =

Explanation / Answer

Let the left vertex be at origin .

lower limit of integration a= 0

upper limit b= 5

2r = f(x) - g(x)

To find f(x) and g(x) cut the base horizontally from the left vertex we get two 30 - 60 - 90 degrees triangle.

f(x) = x/sqrt(3) (tan 30 = 1/sqrt(3) = y/x ==> y = x/sqrt(3) )

Similarly g(x) = -x/sqrt(3)

2r = 2x/sqrt(3)

r = x/sqrt(3)

A(x) = (1/2)pi.r^2 = (1/6) pi .x^2

V = integral from x = 0 to 5 (1/6) pi .x^2dx

V = x = 0 to 5 (1/18) pi .x^3 = 125pi/18

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