Calculating the Ratio of Diameter to Height for a Cylindrical Pillar: A Comprehensive Guide
In this article, we'll thoroughly explore a mathematical problem involving a cylindrical pillar and its surface area and volume. We will pinpoint the pillar's height, diameter, and finally, the ratio of its diameter to its height. Let's break down a typical problem of this kind into manageable steps and mathematical formulas.
Problem Description
The curved surface area of a cylindrical pillar is given as 616 m2 and its volume is 2156 m3. The task is to determine the ratio of its diameter to its height.
First Approach to Solving the Problem
Given Information
Curved Surface Area (C.S.A.): 616 m2 Volume: 2156 m3
Step 1: Use the Given C.A. Formula
The formula for the curved surface area (C.S.A.) of a cylinder is given by:
C.S.A. 2πrh
Given:
2πrh 616
Step 2: Use the Given Volume Formula
The formula for the volume (V) of a cylinder is given by:
V πr2h
Given:
πr2h 2156
Step 3: Relate the Volume and Curved Surface Area
We know that:
πr2h / 2πrh 2156 / 616
After simplifying, we find:
r / 2 1078 / 77 98 / 7 14
Hence, r 14 meters.
Step 4: Substitute r into the Curved Surface Area Formula
Now, substituting r 14 into the C.S.A. formula:
2πrh 616
2π × 14 × h 616
44h 616
H 616 / 44 14 meters
Step 5: Find the Ratio of Diameter to Height
Diameter of the cylinder 2r 2 × 14 28 meters
Ratio of diameter to height 28 / 14 2
Therefore, the ratio of diameter to height is 2:1.
Second Approach to Solving the Problem
Given Information (Rephrasing)
C.S.A. of the cylinder 616 m2 and volume 2156 m3.
Step 1: Simplify Using C.S.A. and Volume
C.S.A. 2πrh 616
Volume πr2h 2156
Dividing the two equations:
r 924 / 264 7 metres
Step 2: Determine the Height
Using the C.S.A. formula again:
2πrh 2π × 7 × h 616
44 × h 616
h 616 / 44 14 metres
Step 3: Calculate the Ratio of Diameter to Height
Diameter 2r 14 × 2 28 metres
Ratio of diameter to height 28 / 14 2:1
Third Approach to Solving the Problem
Given Information (Rephrasing)
C.S.A. of the cylinder 2πrh 616
Volume of the cylinder πr2h 2156
Step 1: Simplify Using C.S.A. and Volume
C.S.A. formula: 2πrh 616
Volume formula: πr2h 2156
Divide the two equations:
r 924 / 264 7 metres
Step 2: Determine the Height
Substitute r 7 into the C.S.A. formula:
2π × 7 × h 616
44h 616
h 616 / 44 14 metres
Step 3: Calculate the Ratio of Diameter to Height
Diameter 2r 2 × 7 14 metres
Ratio of diameter to height 14 / 14 1:1
Therefore, the ratio of diameter to height is 1:1.
Fourth and Fifth Approaches
The mathematical steps in the other examples follow similar logic and ultimately yield the same conclusion that the ratio of diameter to height in this case is 1:1.
This guide has explored multiple approaches to solving a problem involving the diameter-to-height ratio of a cylindrical pillar given its surface area and volume. The key formulas used include the curved surface area (2πrh) and volume (πr2h) of the cylinder. These methods confirm the ratio, ensuring a comprehensive understanding of the underlying principles.