A certain solid S has a horizontal plane region R as its base and has height h cm measured perpendicular to R. For 0

Question

A certain solid S has a horizontal plane region R as its base and has height h cm measured perpendicular to R. For 0 < z < h, the volume of that part of S lying beneath the plane at height z cm above R is V(z) = 2z + z³ cm³. Find (a) the area of the cross-section of S in the plane at height z cm and (b) the area of R. A) (a) 2 + 3z² cm², (b) 2 cm² B) (a) 1 + 3z² cm², (b) 1 cm² C) (a) 3 + z² cm², (b) 3 cm² D) (a) 2 + 4z² cm², (b) 3 cm² E) (a) 1 + 2z² cm², (b) 1 cm²

Quizplus · Accepted Answer

The volume function V(z) = 2z + z^3 represents the volume of the solid up to height z. The derivative of V(z) with respect to z, V'(z) = 2 + 3z^2, gives the rate of change of volume, which is the area of the cross-section at height z. Therefore, the area of the cross-section at height z is 2 + 3z^2. To find the area of R, we set z = 0 in the cross-section area formula, giving an area of 2.

A Certain Solid S Has a Horizontal Plane Region R