Question

The curved surface area of a cylinder is 80 sq m and its total surface area is 140 sq m. Find the ratio of the radius and the height of the cylinder.

Answer 1

To find the ratio of the radius and the height of the cylinder, we can use the formulas for the curved surface area and the total surface area of a cylinder.

1. The formula for the curved surface area (CSA) of a cylinder is given by:
\[
\text{CSA} = 2\pi rh
\]
where \( r \) is the radius and \( h \) is the height.

2. The formula for the total surface area (TSA) of a cylinder is given by:
\[
\text{TSA} = 2\pi r(h + r)
\]

Given:
- Curved Surface Area (CSA) = 80 sq m
- Total Surface Area (TSA) = 140 sq m

From the CSA we have:
\[
2\pi rh = 80
\]
From the TSA we have:
\[
2\pi r(h + r) = 140
\]

Now, we will solve these equations step-by-step.

### Step 1: Solve for \( rh \) from CSA
\[
2\pi rh = 80
\]
Dividing both sides by \( 2\pi \), we get:
\[
rh = \frac{80}{2\pi} = \frac{40}{\pi}
\]

### Step 2: Substitute \( h \) from CSA into TSA
From \( rh = \frac{40}{\pi} \), we can express \( h \) in terms of \( r \):
\[
h = \frac{40}{\pi r}
\]

Now substitute \( h \) in the TSA equation:
\[
2\pi r \left( \frac{40}{\pi r} + r \right) = 140
\]

### Step 3: Simplify the TSA equation
Substituting \( h \):
\[
2\pi r \left( \frac{40}{\pi r} + r \right) = 140
\]
Combining the terms inside the parenthesis:
\[
= 2\pi r \left( \frac{40 + \pi r^2}{\pi r} \right) = 140
\]
This simplifies to:
\[
2(40 + \pi r^2) = 140
\]

Dividing both sides by 2:
\[
40 + \pi r^2 = 70
\]
\[
\pi r^2 = 70 - 40
\]
\[
\pi r^2 = 30
\]
Thus,
\[
r^2 = \frac{30}{\pi}
\]

### Step 4: Find height \( h \)
Now, substitute \( r^2 \) back to find \( h \):
From \( rh = \frac{40}{\pi} \), we use:
\[
h = \frac{40}{\pi r}
\]

But we need \( r \) first. We will find \( r \):
\[
r = \sqrt{\frac{30}{\pi}}
\]

Substituting for \( h \):
\[
h = \frac{40}{\pi \sqrt{\frac{30}{\pi}}}
= \frac{40 \sqrt{\pi}}{\sqrt{30}\pi}
= \frac{40}{\sqrt{30\pi}}
\]

### Step 5: Finding the ratio \( \frac{r}{h} \)
Now we have:
\[
\frac{r}{h} = \frac{\sqrt{\frac{30}{\pi}}}{\frac{40}{\sqrt{30\pi}}}
\]
This simplifies to:
\[
\frac{r}{h} = \frac{\sqrt{30}}{\pi} \cdot \frac{\sqrt{30\pi}}{40} = \frac{30}{40} = \frac{3}{4}
\]

The final ratio of the radius to the height is:
\[
\boxed{\frac{3}{4}}
\]