Question

an artist makes a sculpture by placing a clay sphere on a cylinder filled with clay. the height of the cylinder is 1 cm. after placing the sphere on the cylinder, the bottom of the sphere rests \\( \frac { 1 } { 2 } \\) cm below the top of the cylinder, and \\( 1.4 \mathrm { cm } ^ { 3 } \\) of clay from the cylinder has to be removed. which of the following represents the total volume of the clay sculpture? 2 of 4 questions \\( v = \frac { 4 } { 3 } \pi r ^ { 3 } + \pi - 1.4 \mathrm { cm } ^ { 3 } \\) \\( v = \frac { 4 } { 3 } \pi r ^ { 3 } + 4 \pi + 1.4 \mathrm { cm } ^ { 3 } \\) \\( v = \frac { 4 } { 3 } \pi r ^ { 3 } + \pi + 1.4 \mathrm { cm } ^ { 3 } \\) \\( v = \frac { 4 } { 3 } \pi r ^ { 2 } + \frac { 1 } { 2 } \pi - 1.4 \mathrm { cm } ^ { 3 } \\)

Explanation:

Step1: Calculate the volume of the sphere

The formula for the volume of a sphere is \(V_{sphere}=\frac{4}{3}\pi r^{3}\).

Step2: Calculate the volume of the cylinder

The diameter of the cylinder is \(d = 2\) cm, so the radius \(r_{cylinder}=1\) cm and height \(h = 1\) cm. The volume of a cylinder is \(V_{cylinder}=\pi r^{2}h=\pi\times(1)^{2}\times1=\pi\) \(cm^{3}\).

Step3: Determine the total volume of the sculpture

The total volume of the sculpture is the sum of the volume of the sphere and the volume of the cylinder minus the volume of the clay removed (\(1.4\) \(cm^{3}\)). So \(V = V_{sphere}+V_{cylinder}- 1.4=\frac{4}{3}\pi r^{3}+\pi - 1.4\) \(cm^{3}\)

Answer:

\(V=\frac{4}{3}\pi r^{3}+\pi - 1.4\) \(cm^{3}\) (the first option)