The radius and height of a cylinder are in the ratio `5:7` and its volume is `550cm^(3)`. Find its radius and height.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We know that the radius (r) and height (h) of the cylinder are in the ratio 5:7, and the volume of the cylinder is 550 cm³.

### Step 2: Express radius and height in terms of a variable
Let the radius of the cylinder be \( r = 5x \) and the height be \( h = 7x \), where \( x \) is a common multiplier.

### Step 3: Write the formula for the volume of a cylinder
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
Substituting the values of \( r \) and \( h \):
\[ V = \pi (5x)^2 (7x) \]

### Step 4: Substitute the known volume into the equation
We know the volume is 550 cm³, so we can set up the equation:
\[ 550 = \pi (5x)^2 (7x) \]
Using \( \pi \approx \frac{22}{7} \):
\[ 550 = \frac{22}{7} (5x)^2 (7x) \]

### Step 5: Simplify the equation
Calculating \( (5x)^2 \):
\[ (5x)^2 = 25x^2 \]
Now substitute this back into the volume equation:
\[ 550 = \frac{22}{7} \cdot 25x^2 \cdot 7x \]
The \( 7 \) in the numerator and denominator cancels out:
\[ 550 = 22 \cdot 25x^2 \cdot x \]
This simplifies to:
\[ 550 = 550x^3 \]

### Step 6: Solve for \( x \)
Now, divide both sides by 550:
\[ 1 = x^3 \]
Taking the cube root of both sides:
\[ x = 1 \]

### Step 7: Find the radius and height
Now that we have \( x \), we can find the radius and height:
- Radius \( r = 5x = 5 \cdot 1 = 5 \) cm
- Height \( h = 7x = 7 \cdot 1 = 7 \) cm

### Final Answer
The radius of the cylinder is **5 cm** and the height is **7 cm**.

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