Question

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

Which of these best compares their surface areas and volumes?

Group of answer choices

The prism has a surface area about 19 square centimeters smaller than the cone.


The prism has a surface area about 11 square centimeters larger than the cone.


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


The prism has a volume about 275 cubic centimeters larger than the cone.


Write your response in your journal and upload your answer.

Correctly state yes or no (4 points)
Include a valid explanation (6 points)

Answer 1

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

Given:

▪
`\longrightarrow \sf{V_c = \frac{ {\pi r}^(2) h}{3} }`

▪
`\longrightarrow \sf{Radius= 6cm}`

▪
`\longrightarrow \sf{Height=3cm}`

`\large\leadsto` The volume is:

`\sf\longrightarrow{V_c= \frac{ \pi {3}^(2) \: \cdot \: 6 }{3} = (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= (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

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

Explanation:

Cone

Formulas

`\sf Surface\:area\:of\:a\:cone=\pi r \left(r+√(h^2+r^2)\right)`

`\textsf{Volume of a cone}=\sf (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+√(6^2+3^2)\right)\\ & = 3 \pi \left (3+√(36+9)\right)\\ & = 3\pi (3+√(45))\\ & = 3\pi(3+3√(5))\\ & = 91.5 \:\: \sf cm^2\:(1\:d.p.)\end{aligned}`

`\begin{aligned}\textsf{Volume of cone} & =(1)/(3) \pi (3)^2 (6)\\& = (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} * height`

`\textsf{Area of a triangle}=\sf (1)/(2) * base * height`

`\textsf{Area of a rectangle}=\sf width * 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} & = (1)/(2) * 8 * 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 & = √(164)\\c & = 2√(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 * 10)+(10 * 8)+ (10 * 2√(41))\\& = 80 + 100 + 80 + 20√(41)\\& = 388.1 \:\: \sf cm^2\:(1\:d.p.)\end{aligned}`

`\begin{aligned}\textsf{Volume of prism} & = 40 * 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.