Multiple Choice Question
An engineer is designing a cylinder-shaped structure using the formula shown below. As shown, an approximation of 22/7 for PIE 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 when the volume is an irrational number, we need to find a pair of values for the radius and height that result in an irrational volume.
The formula for the volume of a cylinder is given by V = πr^2h, where V represents the volume, π is a constant (approximated as 22/7), r is the radius, and h is the height.
For the volume to be irrational, both πr^2 and h must be irrational numbers.
Let's examine the given options:
A) r = √2 meters, h = √5 meters:
πr^2 = π(√2)^2 = π(2) = 2π, which is irrational (π is irrational).
h = √5 meters, which is irrational.
Therefore, this option could result in an irrational volume.
B) r = 3 meters, h = √7 meters:
πr^2 = π(3)^2 = 9π, which is irrational (π is irrational).
h = √7 meters, which is irrational.
Therefore, this option could result in an irrational volume.
C) r = 5 meters, h = 2 meters:
πr^2 = π(5)^2 = 25π, which is rational (π is irrational, but 25 is rational).
h = 2 meters, which is rational.
Therefore, this option will result in a rational volume.
D) r = 1 meter, h = π meters:
πr^2 = π(1)^2 = π, which is rational.
h = π meters, which is rational.
Therefore, this option will result in a rational volume.
From the given options, only options A) and B) have the potential to result in an irrational volume.
To determine which pair of values will result in the volume being an irrational number, we need to consider the formula for the volume of a cylinder and evaluate the result.
The formula for the volume of a cylinder is:
V = π * r^2 * h
where V represents the volume, π is a constant (approximated as 22/7), r is the radius, and h is the height of the cylinder.
For the volume to be an irrational number, at least one of the factors on the right side of the equation must be irrational.
Let's consider the options:
A. r = 1/2 m, h = 5 m
Substituting the values into the formula:
V = (22/7) * (1/2)^2 * 5
Calculating this result:
V = (22/7) * (1/4) * 5 = 55/14 m³
The result is a rational number, not an irrational one.
B. r = √2 m, h = 7 m
Substituting the values into the formula:
V = (22/7) * (√2)^2 * 7
Calculating this result:
V = (22/7) * 2 * 7 = 308/7 m³
The result is a rational number, not an irrational one.
C. r = 3/4 m, h = π m
Substituting the values into the formula:
V = (22/7) * (3/4)^2 * π
Calculating this result:
V = (22/7) * (9/16) * π = (99/32) * π m³
Since π is an irrational number, multiplying it by a rational number (99/32) results in an irrational number for the volume. This option makes the volume an irrational number.
Therefore, the pair of values that makes the volume an irrational number is r = 3/4 m and h = π m (option C).
To determine which pair of values will result in an irrational number for the volume of the cylinder, we need to find the volume of the cylinder using the given formula and check if the final result is irrational.
The formula for the volume of a cylinder is:
Volume = π * radius^2 * height
In this case, an approximation of 22/7 is used for π. To determine if the volume will be irrational, we can ignore the approximation and use the exact value of π.
Now, let's consider different pairs of values for the radius (r) and height (h) to find out which one will result in an irrational volume.
Option A: r = 1 meter and h = 1 meter
Volume = π * 1^2 * 1 = π * 1 * 1 = π
Since π is irrational, the volume will be irrational.
Option B: r = 2 meters and h = 3 meters
Volume = π * 2^2 * 3 = π * 4 * 3 = 12π
Since π is irrational, the volume will also be irrational.
Option C: r = 5 meters and h = 7 meters
Volume = π * 5^2 * 7 = π * 25 * 7 = 175π
Once again, since π is irrational, the volume will be irrational.
Therefore, all of the given pairs of values (Option A, Option B, and Option C) will result in an irrational volume for the cylinder.