The radius and height of a cylinder are in the ratio `5 : 7` and its volume is 4400 `cm^3`. Find the unit digit of height.

Answer

To solve the problem step-by-step, we will follow these instructions:

### Step 1: Understand the given information
The radius and height of a cylinder are in the ratio of \(5 : 7\). The volume of the cylinder is given as \(4400 \, \text{cm}^3\).

### Step 2: Express radius and height in terms of a variable
Let the radius be \(r\) and the height be \(h\). According to the ratio:
- Radius \(r = 5x\)
- Height \(h = 7x\)

### Step 3: Write the formula for the volume of a cylinder
The volume \(V\) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Substituting the values of \(r\) and \(h\):
\[ 4400 = \pi (5x)^2 (7x) \]

### Step 4: Substitute the value of \(\pi\)
Using \(\pi \approx \frac{22}{7}\):
\[ 4400 = \frac{22}{7} (5x)^2 (7x) \]

### Step 5: Simplify the equation
Calculating \((5x)^2\):
\[ (5x)^2 = 25x^2 \]
Now substitute this back into the volume equation:
\[ 4400 = \frac{22}{7} \cdot 25x^2 \cdot 7x \]
The \(7\) in the denominator and the \(7\) in the numerator cancel out:
\[ 4400 = 22 \cdot 25x^2 \cdot x \]
This simplifies to:
\[ 4400 = 550x^3 \]

### Step 6: Solve for \(x^3\)
Now, divide both sides by \(550\):
\[ x^3 = \frac{4400}{550} \]
Calculating the right side:
\[ x^3 = 8 \]

### Step 7: Calculate \(x\)
Taking the cube root of both sides:
\[ x = \sqrt[3]{8} = 2 \]

### Step 8: Find the height
Now, substitute \(x\) back to find the height:
\[ h = 7x = 7 \cdot 2 = 14 \, \text{cm} \]

### Step 9: Determine the unit digit of the height
The unit digit of \(14\) is \(4\).

### Final Answer
The unit digit of the height is \(4\).

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