A solid has a base on the region in the –plane bounded by the –axis

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

A solid has a base on the region in the

xy

–plane bounded by the

x

–axis, the graphs of

y=\sqrt{x}

and

y=0

, and the vertical line

x=a

, where

a>0

. For every

x

in this interval, the cross section perpendicular to the

x

–axis is an equilateral triangle whose side lies in the base region. The manufacturing specification requires the volume of the solid to be exactly

60

cubic units. (a) Express the side length

s(x)

of the equilateral triangle at position

x

in terms of

x

. Justify your expression using the geometry of the base region. (b) Write, but do not evaluate, a definite integral that represents the volume of the solid in terms of

a

. Clearly indicate the limits of integration. (c) Evaluate the integral from part (b) to obtain a formula for the volume as a function of

a

, and hence set up an equation whose solution gives the value of

a

that produces the required volume of

60

cubic units. (d) Solve the equation from part (c) for

a

. Show all algebraic steps and state the final value of

a

correct to three decimal places.

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