The total surface area of a cylinder is `368 pi cm ^(2).` and sum of radius and height of cylinder is `23cm.` Find the volume of cone whose total surface area is `200 pi cm ^(2).` (radius of cylinder and cone is equal)

Options

A. A)512`pi` `cm^(3)`
B. B)640`pi` `cm^(3)`
C. C)320`pi` `cm^(3)`
D. D)290`pi` `cm^(3)`

Answer

To solve the problem step by step, we will follow the given information and apply the relevant formulas.

### Step 1: Understand the Total Surface Area of the Cylinder
The total surface area (TSA) of a cylinder is given by the formula:
\[ \text{TSA} = 2\pi r(h + r) \]
We know that the TSA of the cylinder is \(368\pi\) cm². Therefore, we can set up the equation:
\[ 2\pi r(h + r) = 368\pi \]

### Step 2: Simplify the Equation
We can divide both sides of the equation by \(\pi\):
\[ 2r(h + r) = 368 \]
Next, divide by 2:
\[ r(h + r) = 184 \]

### Step 3: Use the Given Sum of Radius and Height
We are also given that the sum of the radius and height of the cylinder is \(23\) cm:
\[ h + r = 23 \]
From this, we can express \(h\) in terms of \(r\):
\[ h = 23 - r \]

### Step 4: Substitute \(h\) in the TSA Equation
Now, substitute \(h\) into the equation \(r(h + r) = 184\):
\[ r(23) = 184 \]
This simplifies to:
\[ 23r = 184 \]

### Step 5: Solve for the Radius \(r\)
Now, divide both sides by \(23\):
\[ r = \frac{184}{23} = 8 \text{ cm} \]

### Step 6: Find the Height \(h\)
Using the value of \(r\) in the equation \(h = 23 - r\):
\[ h = 23 - 8 = 15 \text{ cm} \]

### Step 7: Find the Total Surface Area of the Cone
The total surface area (TSA) of a cone is given by the formula:
\[ \text{TSA} = \pi r(L + r) \]
We know the TSA of the cone is \(200\pi\) cm². Therefore, we can set up the equation:
\[ \pi r(L + r) = 200\pi \]
Dividing both sides by \(\pi\):
\[ r(L + r) = 200 \]

### Step 8: Substitute the Radius of the Cone
Since the radius of the cone is equal to the radius of the cylinder, we substitute \(r = 8\) cm:
\[ 8(L + 8) = 200 \]

### Step 9: Solve for the Slant Height \(L\)
Now, divide both sides by \(8\):
\[ L + 8 = 25 \]
Subtract \(8\) from both sides:
\[ L = 17 \text{ cm} \]

### Step 10: Find the Height of the Cone
To find the height \(h_c\) of the cone, we use the Pythagorean theorem:
\[ h_c = \sqrt{L^2 - r^2} \]
Substituting the values:
\[ h_c = \sqrt{17^2 - 8^2} = \sqrt{289 - 64} = \sqrt{225} = 15 \text{ cm} \]

### Step 11: Calculate the Volume of the Cone
The volume \(V\) of the cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h_c \]
Substituting the values:
\[ V = \frac{1}{3} \pi (8^2)(15) = \frac{1}{3} \pi (64)(15) = \frac{960}{3} \pi = 320 \pi \text{ cm}^3 \]

### Final Answer
The volume of the cone is \(320 \pi \text{ cm}^3\).

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