Cylinder Ratio Calculation: Height to Diameter
We are given the curved surface area (CSA) and volume (V) of a cylindrical pillar and asked to find the ratio of its height ($h$) to its diameter ($d$).
Given:
- CSA = $264 \text{ m}^2$
- V = $924 \text{ m}^3$
Formulas for a cylinder:
- CSA = $2 \pi r h$
- V = $\pi r^2 h$
- Diameter $d = 2r$
Here, $r$ is the radius and $h$ is the height.
Finding Cylinder Dimensions
Step 1: Relate Volume and Surface Area
Divide the volume formula by the curved surface area formula:
$ \frac{V}{CSA} = \frac{\pi r^2 h}{2 \pi r h} $Simplify the expression:
$ \frac{V}{CSA} = \frac{r}{2} $Substitute the given values:
$ \frac{924}{264} = \frac{r}{2} $Simplify the fraction $\frac{924}{264}$:
$ \frac{924}{264} = \frac{7 \times 132}{2 \times 132} = \frac{7}{2} $Therefore,
$ \frac{r}{2} = \frac{7}{2} $This gives us the radius $r = 7$ meters.
Step 2: Calculate the Height
Use the curved surface area formula $CSA = 2 \pi r h$ and substitute the known values ($CSA = 264$, $r = 7$). Use $\pi \approx \frac{22}{7}$.
$ 264 = 2 \times \frac{22}{7} \times 7 \times h $ $ 264 = 44 h $Solve for $h$:
$ h = \frac{264}{44} $ $ h = 6 \text{ meters} $Determining the Ratio
Step 3: Calculate the Diameter
The diameter $d$ is twice the radius:
$ d = 2r = 2 \times 7 = 14 \text{ meters} $Step 4: Find the Ratio of Height to Diameter
Calculate the ratio $h:d$:
$ \frac{h}{d} = \frac{6}{14} $Simplify the ratio:
$ \frac{h}{d} = \frac{3}{7} $The ratio of the height to the diameter is $3:7$.
Similar Questions
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- The total surface area of a cube having side 40 cm is:
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Important Questions from 3-D Mensuration
- Find the surface area of a sphere whose diameter is equal to 98 cm.
- A conical vessel has base radius 31 cm and height 45 cm. Water is poured into the vessel until it is $\frac{2}{3}$ full. Find the volume (in cm³) of water in the vessel.
- A solid metallic sphere of radius 10 cm is melted and recast into 125 identical spheres. What is the ratio of the surface area of the original sphere to the total surface area of 6 smaller spheres so formed?
- Find the volume (in cm³) of the largest right circular cone that can be cut out from a cube with an edge of 8 cm. Use $\pi = \frac{22}{7}$
- The volume of a solid cylinder is 5852 cm³ and its height is 38 cm. What is the total surface area of the solid cylinder? (Round your answer to the nearest integer) (Use $\pi = \frac{22}{7}$)
