Question

determine the volume and the surface area of the three - dimensional figure.
use the π key on your calculator.
a) the volume is 29.32 cm³.
(simplify your answer. type an integer or a decimal rounded to the nearest
hundredth as needed.)
b) the surface area is □ cm².
(simplify your answer. type an integer or a decimal rounded to the nearest
hundredth as needed.)

Explanation:

Step1: Find the height of the cone

The total length is \(7\) cm and the radius \(r = 2\) cm. Let the height of the cone be \(h\). Using the Pythagorean theorem for the cone (slant - height is not needed directly here, but we know that for the composite figure, if the cylinder has height \(h_{cylinder}\) and cone has height \(h_{cone}\), and assuming the cylinder's height \(h_{cylinder}=r = 2\) cm (since it's a right - circular cylinder and cone combination in a simple case where the non - overlapping part for height: \(h_{cone}=7 - 2=5\) cm.

Step2: Calculate the surface area of the composite figure

The surface area of the composite figure \(S\) is the sum of the lateral surface area of the cone \(S_{cone}\) and the lateral surface area of the cylinder \(S_{cylinder}\) plus the area of the circular base of the cylinder (since the cone has no base in the composite figure).
The formula for the lateral surface area of a cone is \(S_{cone}=\pi rl\), where \(l=\sqrt{h^{2}+r^{2}}\) (slant height). \(l = \sqrt{5^{2}+2^{2}}=\sqrt{25 + 4}=\sqrt{29}\approx5.39\) cm. \(S_{cone}=\pi\times2\times\sqrt{29}\)
The formula for the lateral surface area of a cylinder is \(S_{cylinder}=2\pi rh_{cylinder}\), with \(h_{cylinder}=2\) cm. \(S_{cylinder}=2\pi\times2\times2 = 8\pi\)
The area of the base of the cylinder is \(A=\pi r^{2}=\pi\times2^{2}=4\pi\)
\(S=\pi\times2\times\sqrt{29}+8\pi + 4\pi=\pi(2\sqrt{29}+12)\)
\(S\approx3.14\times(2\times5.39+12)=3.14\times(10.78 + 12)=3.14\times22.78\approx71.53\)

Answer:

a) \(29.32\) (already given)
b) \(71.53\)