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#mathematics
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#geometry
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The total surface area of a cone is 1296πcm and it's base radius is 16cm find the slant height and the volume ?
Ijeoma Ukanze
more than 1 week ago
The total surface area of a cone is 1296πcm and it's base radius is 16cm find the slant height and the volume ?
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ChatGPT
more than 1 week ago
To find the slant height of the cone, we can use the formula for the total surface area of a cone:
Total Surface Area = πr(r + l)
Given that the total surface area is 1296π cm and the base radius is 16 cm, we can plug in these values and solve for the slant height (l):
1296π = π(16)(16 + l)
1296 = 16(16 + l)
81 = 16 + l
l = 65
Therefore, the slant height of the cone is 65 cm.
To find the volume of the cone, we can use the formula:
Volume = (1/3)πr^2h
Given that the base radius is 16 cm and the slant height is 65 cm, we can plug in these values and solve for the volume (h):
Volume = (1/3)π(16)^2(65)
Volume = (1/3)π(256)(65)
Volume = (1/3)π(16640)
Volume = 5546.67π
Therefore, the volume of the cone is 5546.67π cm^3.
Total Surface Area = πr(r + l)
Given that the total surface area is 1296π cm and the base radius is 16 cm, we can plug in these values and solve for the slant height (l):
1296π = π(16)(16 + l)
1296 = 16(16 + l)
81 = 16 + l
l = 65
Therefore, the slant height of the cone is 65 cm.
To find the volume of the cone, we can use the formula:
Volume = (1/3)πr^2h
Given that the base radius is 16 cm and the slant height is 65 cm, we can plug in these values and solve for the volume (h):
Volume = (1/3)π(16)^2(65)
Volume = (1/3)π(256)(65)
Volume = (1/3)π(16640)
Volume = 5546.67π
Therefore, the volume of the cone is 5546.67π cm^3.
0
Google Gemini AI
more than 1 week ago
Step 1: Find the height of the cone
The total surface area of a cone is given by the formula:
```
TSA = πr^2 + πrl
```
where:
- TSA is the total surface area
- r is the radius of the base
- l is the slant height
We are given that the total surface area is 1296πcm and the radius of the base is 16cm. Plugging these values into the formula, we get:
```
1296πcm = π(16cm)^2 + π(16cm)l
```
Simplifying the equation, we get:
```
1296πcm = 256πcm + 16πcm - l
```
Subtracting 256πcm from both sides, we get:
```
1140πcm = 16πcm - l
```
Dividing both sides by 16πcm, we get:
```
l = 71cm
```
Therefore, the slant height of the cone is 71cm.
Step 2: Find the volume of the cone
The volume of a cone is given by the formula:
```
Volume = (1/3)πr^2h
```
where:
- Volume is the volume of the cone
- r is the radius of the base
- h is the height of the cone
We know the radius of the base is 16cm and the height of the cone is 71cm. Plugging these values into the formula, we get:
```
Volume = (1/3)π(16cm)^2(71cm)
```
Simplifying the equation, we get:
```
Volume = (1/3)π(256cm)(71cm)
```
Multiplying the numbers, we get:
```
Volume = 5776πcm^3
```
Therefore, the volume of the cone is 5776πcm^3.
The total surface area of a cone is given by the formula:
```
TSA = πr^2 + πrl
```
where:
- TSA is the total surface area
- r is the radius of the base
- l is the slant height
We are given that the total surface area is 1296πcm and the radius of the base is 16cm. Plugging these values into the formula, we get:
```
1296πcm = π(16cm)^2 + π(16cm)l
```
Simplifying the equation, we get:
```
1296πcm = 256πcm + 16πcm - l
```
Subtracting 256πcm from both sides, we get:
```
1140πcm = 16πcm - l
```
Dividing both sides by 16πcm, we get:
```
l = 71cm
```
Therefore, the slant height of the cone is 71cm.
Step 2: Find the volume of the cone
The volume of a cone is given by the formula:
```
Volume = (1/3)πr^2h
```
where:
- Volume is the volume of the cone
- r is the radius of the base
- h is the height of the cone
We know the radius of the base is 16cm and the height of the cone is 71cm. Plugging these values into the formula, we get:
```
Volume = (1/3)π(16cm)^2(71cm)
```
Simplifying the equation, we get:
```
Volume = (1/3)π(256cm)(71cm)
```
Multiplying the numbers, we get:
```
Volume = 5776πcm^3
```
Therefore, the volume of the cone is 5776πcm^3.
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