Technology
Understanding and Calculating the Radius of a Compound Solid with a Cone and Cylinder
Understanding and Calculating the Radius of a Compound Solid with a Cone and Cylinder
Introduction
When dealing with compound solids in geometry, such as a solid cone placed over a solid cylinder with the same base radius, it becomes crucial to understand how to calculate the radius of these shapes. This article will guide you through the process of determining the radius of a cone and a cylinder when the total volume is given.
Given Problem
The problem at hand is as follows: A solid cone with a height of 2 meters is placed over a solid cylinder with a height of 3 meters, both having the same base radius. The total volume of both of them is 10,364 cubic meters (M3). The formula for the total volume is given by:
Total Volume πr2hcone/3 πr2hcylinder
Step-by-Step Solution
To find the radius r considering the given volume, we start by understanding the formula and apply it to the given values. The volume of the cone and cylinder can be expressed as follows:
Volume of Cone frac{1}{3}πr2hcone
Volume of Cylinder πr2hcylinder
Where:
hcone 2 meters hcylinder 3 meters Total Volume 10,364 M3Combining the two formulas for the volume, we get:
Total Volume frac{1}{3}πr22 πr23
This simplifies to:
10,364 frac{1}{3}πr22 πr23
Simplifying the Equation
We can factor out πr2 from the equation:
10,364 frac{1}{3}πr22 πr23
10,364 πr2(frac{1}{3}2 3)
10,364 πr2(0.6667 3)
10,364 πr2(3.6667)
To isolate r2, we divide both sides by 3.6667π:
r2 10,364 / (3.6667π)
Let's calculate the value of r2 using the exact value of π (3.14159265359):
r2 10,364 / (3.6667 * 3.14159265359)
r2 10,364 / 11.520
r2 899.7174
Taking the square root of both sides gives us:
r √899.7174
r ≈ 30 meters (approx)
Conclusion
Using the given total volume and by following the derived mathematical steps, we have determined that the radius of the base of the cone and cylinder is approximately 30 meters. This result is subject to rounding and can be adjusted depending on the required level of precision.