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
The formula for the volume of a cylinder is given by V = πr^2h.
To determine which pair of values make the volume an irrational number, we need to calculate the volume for each option and check if it results in an irrational number.
Let's calculate the volume for each option:
A. r = 3 and h = 6.54
V = π(3)^2(6.54) = 183.26π
B. r = √11 and h = 6.54
V = π(√11)^2(6.54) = 71.987π
C. r = √11 and h = √9
V = π(√11)^2√9 = 99π
D. r = 3 and h = √15
V = π(3)^2√15 = 45π√15
None of these options result in the volume being an irrational number. Therefore, none of the given pairs of values make the volume an irrational number.
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 given by: V = πr^2h.
Substituting the given values:
A. r = 3 and h = 6.54
V = π(3^2)(6.54)
V = π(9)(6.54)
V = 56.74614...
The volume in option A is not an irrational number.
B. r = √11 and h = 6.54
V = π(√11^2)(6.54)
V = π(11)(6.54)
V = 227.43006...
The volume in option B is not an irrational number.
C. r = √11 and h = √9
V = π(√11^2)(√9)
V = π(11)(3)
V = 33π
The volume in option C is an irrational number as it contains the irrational number π.
D. r = 3 and h = √15
V = π(3^2)(√15)
V = π(9)(√15)
V = 9π√15
The volume in option D is not an irrational number.
Therefore, the pair of values that makes the volume an irrational number is C. r = √11 and h = √9.
To determine which pair of values makes the volume an irrational number, let's analyze the given options one by one.
The formula for the volume of a cylinder is V = πr^2h, where r represents the radius and h represents the height.
A. For r = 3 and h = 6.54:
V = π(3^2)(6.54) = π(9)(6.54) = 186.684...
The volume is not an irrational number; it is a real number since π is a transcendental number.
B. For r = √11 and h = 6.54:
V = π(√11)^2(6.54) = π(11)(6.54) = 227.409...
The volume is not an irrational number; it is a real number since √11 is an irrational number.
C. For r = √11 and h = √9:
V = π(√11)^2(√9) = π(11)(3) = 33π
The volume is not an irrational number; it is a real number since π is a transcendental number.
D. For r = 3 and h = √15:
V = π(3^2)(√15) = π(9)(√15) = 9π√15
The volume is not an irrational number; it is a real number since π is a transcendental number.
None of the given options make the volume an irrational number. Thus, the answer is none of the above (E).