A cylinder has a surface area of `360pi` and height of 3. What is the diameter of the cylinder's circular base?

Answer

To find the diameter of the cylinder's circular base given the surface area and height, we can follow these steps:

### Step 1: Understand the surface area formula for a cylinder
The total surface area (TSA) of a cylinder is given by the formula:
\[ \text{TSA} = 2\pi r^2 + 2\pi rh \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

### Step 2: Substitute the known values into the formula
We know the total surface area is \( 360\pi \) and the height \( h \) is 3. Substituting these values into the formula gives:
\[ 360\pi = 2\pi r^2 + 2\pi r(3) \]

### Step 3: Simplify the equation
First, we can divide the entire equation by \( \pi \) to simplify:
\[ 360 = 2r^2 + 6r \]

### Step 4: Rearrange the equation
Rearranging gives us a standard quadratic equation:
\[ 2r^2 + 6r - 360 = 0 \]

### Step 5: Divide the equation by 2
To make the calculations easier, divide the entire equation by 2:
\[ r^2 + 3r - 180 = 0 \]

### Step 6: Factor the quadratic equation
We need to factor the quadratic equation \( r^2 + 3r - 180 = 0 \). We look for two numbers that multiply to \(-180\) and add to \(3\). The numbers \(15\) and \(-12\) work:
\[ (r + 15)(r - 12) = 0 \]

### Step 7: Solve for \( r \)
Setting each factor equal to zero gives us:
1. \( r + 15 = 0 \) → \( r = -15 \) (not valid since radius cannot be negative)
2. \( r - 12 = 0 \) → \( r = 12 \)

### Step 8: Calculate the diameter
The diameter \( d \) of the circular base is twice the radius:
\[ d = 2r = 2 \times 12 = 24 \]

### Final Answer
The diameter of the cylinder's circular base is \( 24 \).

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