Multiple Choice Question
An engineer is designing a cylinder-shaped structure using the formula shown below. As shown, an approximation of 22/7 for 3.15 is used in the formula.
The engineer is considering using different values for , the radius in meters, and , the height in meters. Which pair of values makes , the volume in cubic meters, an irrational number?
To determine which pair of values makes the volume an irrational number, we need to consider the value of π (pi) in the formula.
The formula for the volume of a cylinder is given by: V = πr^2h
Since the engineer is using an approximation of 22/7 for π, the volume will only be an irrational number if the values of r and h result in an irrational value for V when multiplied by π.
Therefore, we can conclude that the pair of values that makes the volume an irrational number is not provided in the given information. More information is needed about the values of r and h.
To determine which pair of values for the radius and height makes the volume an irrational number, we need to find a combination that results in a non-terminating and non-repeating decimal when calculating the volume.
The formula for the volume of a cylinder is given by:
Volume = π * (radius^2) * height
Since the engineer is using an approximation of 22/7 for π, we can rewrite the volume formula as:
Volume ≈ (22/7) * (radius^2) * height
In order for the volume to be an irrational number, we need to find values for the radius and height such that the calculated volume is an irrational number.
Let's consider the options:
a) radius = 1 meter, height = 3 meters:
Volume ≈ (22/7) * (1^2) * 3
= (22/7) * 1 * 3
= 66/7
≈ 9.42857
The volume is a rational number, not an irrational number.
b) radius = 3 meters, height = 1 meter:
Volume ≈ (22/7) * (3^2) * 1
= (22/7) * 9 * 1
= 198/7
≈ 28.28571
Again, the volume is a rational number, not an irrational number.
c) radius = 2 meters, height = 2 meters:
Volume ≈ (22/7) * (2^2) * 2
= (22/7) * 4 * 2
= 176/7
= 25.14286
Once more, the volume is a rational number, not an irrational number.
d) radius = 5 meters, height = 1 meter:
Volume ≈ (22/7) * (5^2) * 1
= (22/7) * 25 * 1
= 550/7
≈ 78.57143
The volume is still a rational number, not an irrational number.
None of the given pairs of values result in a volume that is an irrational number.
To determine which pair of values makes the volume an irrational number, we need to understand the properties of irrational numbers. An irrational number is a number that cannot be expressed as a fraction of two integers and has a non-repeating decimal expansion.
Let's start by looking at the formula for the volume of a cylinder:
Volume = π * radius^2 * height
Since we are given an approximation of 22/7 for π, we can rewrite the formula as:
Volume ≈ (22/7) * radius^2 * height
To determine if the volume is an irrational number, we need to examine the values of radius and height. An irrational number can only be obtained if either the radius or the height (or both) is irrational.
Since we are looking for a pair of values, we need to find one irrational value for radius and one irrational value for height.
There are many irrational values we can choose from, but for simplicity, let's choose one irrational value for each. For example, we can choose radius = √2 and height = π.
Using these values in the formula:
Volume ≈ (22/7) * (√2)^2 * π
We simplify:
Volume ≈ (22/7) * 2 * π
Volume ≈ (22/7) * 2 * (22/7)
Volume ≈ 4 * 22
Volume ≈ 88
Since the volume obtained is a rational number (88), this pair of values (radius = √2 and height = π) does not make the volume an irrational number.
To find a pair of values that makes the volume an irrational number, we would need to try different irrational values for both the radius and the height.