Question
The curved surface area of a cylinder is 192pi sq cm. If its height and radius are in the ratio 2:3, find its height.
Answers
GPT-4o mini
The curved surface area \( A \) of a cylinder is calculated using the formula:
\[
A = 2\pi r h
\]
where \( r \) is the radius and \( h \) is the height of the cylinder.
Given that the curved surface area is \( 192\pi \) sq cm, we can set up the equation:
\[
2\pi r h = 192\pi
\]
Dividing both sides by \( 2\pi \):
\[
r h = 96
\]
We are also given that the height and radius are in the ratio \( 2:3 \). This means we can express height and radius in terms of a common variable. Let:
\[
h = 2x \quad \text{and} \quad r = 3x
\]
Substituting these expressions into the equation \( r h = 96 \):
\[
(3x)(2x) = 96
\]
This simplifies to:
\[
6x^2 = 96
\]
Dividing both sides by 6:
\[
x^2 = 16
\]
Taking the square root of both sides:
\[
x = 4
\]
Now, we can find the height:
\[
h = 2x = 2 \cdot 4 = 8 \text{ cm}
\]
Therefore, the height of the cylinder is:
\[
\boxed{8 \text{ cm}}
\]
\[
A = 2\pi r h
\]
where \( r \) is the radius and \( h \) is the height of the cylinder.
Given that the curved surface area is \( 192\pi \) sq cm, we can set up the equation:
\[
2\pi r h = 192\pi
\]
Dividing both sides by \( 2\pi \):
\[
r h = 96
\]
We are also given that the height and radius are in the ratio \( 2:3 \). This means we can express height and radius in terms of a common variable. Let:
\[
h = 2x \quad \text{and} \quad r = 3x
\]
Substituting these expressions into the equation \( r h = 96 \):
\[
(3x)(2x) = 96
\]
This simplifies to:
\[
6x^2 = 96
\]
Dividing both sides by 6:
\[
x^2 = 16
\]
Taking the square root of both sides:
\[
x = 4
\]
Now, we can find the height:
\[
h = 2x = 2 \cdot 4 = 8 \text{ cm}
\]
Therefore, the height of the cylinder is:
\[
\boxed{8 \text{ cm}}
\]