A flat, rectangular sheet in the –plane occupies the set and . Two solids are built…

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

Problem

A flat, rectangular sheet in the

xy

–plane occupies the set

0\le x\le 5

and

0\le y\le 3

. Two solids are built on this common base. • Solid A has cross sections perpendicular to the

y

–axis that are equilateral triangles whose bases lie in the plane of the rectangle. • Solid B has cross sections perpendicular to the

y

–axis that are semicircles whose diameters lie in the plane of the rectangle. Both solids are made of the same material of uniform density. (a) Express the volume of each solid as a definite integral in terms of

x

y

, labeling any constants of proportionality that arise from the geometry of the cross sections. (b) Without evaluating the integrals from part (a), determine the exact ratio

\dfrac{V_{A}}{V_{B}}

of the volumes of Solid A to Solid B. (c) Interpret the ratio found in part (b) in the context of placing the two solids on a balance scale. Explain whether the scale would balance if equal lengths of solid A and solid B were cut from the same end of each solid.

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