An engineer is designing a cylinder-shaped structure using the formula shown below. As shown, an approximation of
for 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?
A. r = 3 and h= 6.54
B. r = √11 and h = 6.54
C. r = √11 and h = √9
D. r = 3 and h = √15
To determine which pair of values makes the volume an irrational number, we need to substitute the values into the formula and simplify. The formula for the volume of a cylinder is V = πr^2h.
Option A: r = 3 and h = 6.54
V = π(3^2)(6.54) = 57.93π
Option B: r = √11 and h = 6.54
V = π(√11)^2(6.54) = 71.19π
Option C: r = √11 and h = √9
V = π(√11)^2(√9) = 33π
Option D: r = 3 and h = √15
V = π(3^2)(√15) = 45π√15
Since all the options include π and a square root (√) term, we need to determine if the resulting value is irrational. An irrational number is any real number that cannot be expressed as a fraction, and it has an infinite and non-repeating decimal representation.
Option A: 57.93π is an irrational number since π is irrational.
Option B: 71.19π is an irrational number since π is irrational.
Option C: 33π is an irrational number since π is irrational.
Option D: 45π√15 may or may not be irrational, depending on whether √15 is irrational. Simplifying √15 gives us the value of approximately 3.873, which is irrational since it cannot be expressed as a fraction.
Therefore, the pair of values that makes the volume an irrational number is r = 3 and h = √15. Answer: D.
To determine which pair of values will make the volume an irrational number, we need to evaluate the formula for each option and check if the result is an irrational number.
The formula for the volume of a cylinder is given by V = πr²h, where r is the radius of the cylinder and h is the height.
Let's evaluate the formula for each option:
A. r = 3 and h = 6.54
V = π(3)²(6.54)
V ≈ 571.156 ≈ an irrational number
B. r = √11 and h = 6.54
V = π(√11)²(6.54)
V ≈ 537.251 ≈ a rational number, not an irrational number
C. r = √11 and h = √9
V = π(√11)²(√9)
V = π(11)(3)
V ≈ 103.672 ≈ a rational number, not an irrational number
D. r = 3 and h = √15
V = π(3)²(√15)
V ≈ 291.548 ≈ a rational number, not an irrational number
Based on the above evaluations, the pair of values that makes the volume an irrational number is Option A: r = 3 and h = 6.54.
To determine which pair of values for radius (r) and height (h) would result in an irrational volume for a cylinder, we need to consider the formula for the volume of a cylinder.
The formula for the volume of a cylinder is:
V = π * r^2 * h
In order for the volume (V) to be irrational, at least one of the factors must be irrational. Let's examine each option:
A. r = 3 and h = 6.54
In this case, both values are rational, so the volume will be rational.
B. r = √11 and h = 6.54
Here, the radius is the square root of 11, which is an irrational number. Therefore, the volume will be irrational.
C. r = √11 and h = √9
In this case, both the radius (√11) and height (√9) are irrational numbers. Therefore, the volume will be irrational.
D. r = 3 and h = √15
The radius (3) is a rational number, but the height (√15) is an irrational number. Therefore, the volume will be irrational.
So the pairs of values that make the volume (V) an irrational number are:
B. r = √11 and h = 6.54
C. r = √11 and h = √9
D. r = 3 and h = √15