A metal sheet with volump $$ 1080 \mathrm{~cm}^{3} $$ is melted and recast into sma let cones of cqual si Ench cone has base circumference of 12 cm and slant height of 5 cm . The volume of the cone of radlus r and helght h is $$ \mathrm{V}=\frac{3}{3} \pi \mathrm{r}^{2} \mathrm{~h} $$. (a) Calculate the volume of the cone

Question

UpStudy AI · Accepted Answer

To calculate the volume of each cone, let's follow these steps:

### Given:
- **Volume of metal sheet:** $$ 1080 \, \text{cm}^3 $$ (this is used for determining the number of cones, not directly needed for part (a)).
- **Base circumference of the cone:** $$ C = 12 \, \text{cm} $$
- **Slant height of the cone:** $$ l = 5 \, \text{cm} $$
- **Volume formula for a cone:** $$ V = \frac{1}{3} \pi r^2 h $$

### Step 1: Find the Radius ($$ r $$) of the Cone
The circumference $$ C $$ of the base is related to the radius $$ r $$ by the formula:
$$
C = 2\pi r
$$
Solving for $$ r $$:
$$
r = \frac{C}{2\pi} = \frac{12}{2\pi} = \frac{6}{\pi} \approx 1.90986 \, \text{cm}
$$

### Step 2: Find the Height ($$ h $$) of the Cone
Using the Pythagorean theorem for the right triangle formed by the radius, height, and slant height:
$$
l^2 = r^2 + h^2
$$
Solving for $$ h $$:
$$
h = \sqrt{l^2 - r^2} = \sqrt{5^2 - \left(\frac{6}{\pi}\right)^2} = \sqrt{25 - \frac{36}{\pi^2}} \approx \sqrt{25 - 3.6476} \approx \sqrt{21.3524} \approx 4.62 \, \text{cm}
$$

### Step 3: Calculate the Volume ($$ V $$) of the Cone
Now, plug the values of $$ r $$ and $$ h $$ into the volume formula:
$$
V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi \left(\frac{6}{\pi}\right)^2 \times 4.62
$$
Simplifying:
$$
V = \frac{1}{3} \pi \times \frac{36}{\pi^2} \times 4.62 = \frac{12}{\pi} \times 4.62 \approx \frac{12 \times 4.62}{3.1416} \approx \frac{55.44}{3.1416} \approx 17.68 \, \text{cm}^3
$$

### **Final Answer:**
The volume of each cone is approximately **17.7 cm³**.
The volume of each cone is approximately 17.7 cm³.

A metal sheet with volump is melted and recast into sma let cones of cqual si
Ench cone has base circumference of 12 cm and slant height of 5 cm .
The volume of the cone of radlus r and helght h is .
(a) Calculate the volume of the cone

Upstudy AI Solution

Answer

Solution

Given:

Step 1: Find the Radius () of the Cone

Step 2: Find the Height () of the Cone

Step 3: Calculate the Volume () of the Cone

Final Answer:

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A metal sheet with volump is melted and recast into sma let cones of cqual si Ench cone has base circumference of 12 cm and slant height of 5 cm . The volume of the cone of radlus r and helght h is . (a) Calculate the volume of the cone