Question

Consider the right cone and right triangular prism below. Suppose that all measurements are labeled in centimeters.

Answer 1

$\huge\underline{\underline{\boxed{\mathbb {SOLUTION:}}}}$

Given:

▪ $\longrightarrow \sf{V_c = \dfrac{ {\pi r}^{2} h}{3} }$

▪ $\longrightarrow \sf{Radius= 6cm}$

▪ $\longrightarrow \sf{Height=3cm}$

$\large\leadsto$ The volume is:

$\sf\longrightarrow{V_c= \dfrac{ \pi {3}^{2} \: \cdot \: 6 }{3} = \dfrac{54\pi}{3} \approx56.5 {cm}^{3} }$

$\longrightarrow \sf{V_p= BH}$

$\leadsto$ The base is a triangle with a height of 10 cm and a base of 8 cm:

$\longrightarrow \sf{B= \dfrac{10cm \: \cdot \: 8cm}{2} = 40 {cm}^{2} }$

$\leadsto$ The height of the prism is H = 10 cm. Calculate the volume:

$\longrightarrow \sf{V_p= 40cm \: \cdot \: 10cm = 400 {cm}^{3} }$

$\leadsto$ The difference in the volumes is:

$\longrightarrow \sf{400 - 56.5 = 343.5}$

$\huge\underline{\underline{\boxed{\mathbb {ANSWER:}}}}$

$\bm{\small{ The \: prism \: has \: a \: volume \: about \: 340 \: cubic }}$ $\small\bm{\: centimeters \: larger \: than \: the \: cone.}$

Answer 2

Answer:

The prism has a volume about 340 cubic centimeters larger than the cone.

Step-by-step explanation:

Cone

Formulas

$\sf Surface\:area\:of\:a\:cone=\pi r \left(r+\sqrt{h^2+r^2}\right)$

$\textsf{Volume of a cone}=\sf \dfrac{1}{3} \pi r^2 h$

where:

• r = radius of circular base
• h = height perpendicular to the base

Given:

• r = 3 cm
• h = 6 cm

Substitute the given values into the formulas:

$\begin{aligned}\sf Surface\:area\:of\:cone & =\pi (3) \left(3+\sqrt{6^2+3^2}\right)\\ & = 3 \pi \left (3+\sqrt{36+9}\right)\\ & = 3\pi (3+\sqrt{45})\\ & = 3\pi(3+3\sqrt{5})\\ & = 91.5 \:\: \sf cm^2\:(1\:d.p.)\end{aligned}$

$\begin{aligned}\textsf{Volume of cone} & =\dfrac{1}{3} \pi (3)^2 (6)\\& = \dfrac{54}{3} \pi \\ & = 18 \pi \\ & = 56.5\:\: \sf cm^3 \:(1 \: d.p.)\end{aligned}$

Prism

Formulas

$\textsf{Surface area of a prism}=\textsf{Total area of all the sides}$

$\textsf{Volume of a prism}=\sf \textsf{Area of base} \times height$

$\textsf{Area of a triangle}=\sf \dfrac{1}{2} \times base \times height$

$\textsf{Area of a rectangle}=\sf width \times length$

Given:

• Height of triangular base = 10 cm
• Base of triangular base = 8 cm
• Height of prism = 10 cm

Find the area of the triangular base of the prism:

$\begin{aligned}\textsf{Area of the base} & = \dfrac{1}{2} \times 8 \times 10\\& = 40\:\: \sf cm^2\end{aligned}$

Find the third edge of the triangular base by using Pythagoras Theorem:

$\begin{aligned}a^2+b^2 & = c^2\\\implies 8^2+10^2 & = c^2\\164 & = c^2\\c & = \sqrt{164}\\c & = 2\sqrt{41}\end{aligned}$

Use the found values and the formulas to find the surface area of volume of the prism:

$\begin{aligned}\textsf{Surface area of prism} & = \sf 2\:triangles+3\:rectangles\\& = 2\left(40\right) + (10 \times 10)+(10 \times 8)+ (10 \times 2\sqrt{41})\\& = 80 + 100 + 80 + 20\sqrt{41}\\& = 388.1 \:\: \sf cm^2\:(1\:d.p.)\end{aligned}$

$\begin{aligned}\textsf{Volume of prism} & = 40 \times 10\\& = 400\:\:\sf cm^3 \:(1 \:d.p.)\end{aligned}$

Conclusion

The surface area and volume of the prism is larger than that of the cone.

Difference between surface areas:

388.1 - 91.5 = 296.6 ≈ 300 cm²

Difference between volumes:

400 - 56.5 = 343.5 ≈ 340 cm³

Therefore:

• The prism has a surface area about 300 square centimeters larger than the cone.
• The prism has a volume about 340 cubic centimeters larger than the cone.