An engineer is designing a cylinder-shaped structure using the formula shown below. As shown, an approximation of 22/7 for π is used in the formula.
V=22/7r^2h
The engineer is considering using different values for r , the radius in meters, and h , the height in meters. Which pair of values makes V , 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 if the volume V is irrational, we need to check if the values of r, h, and π cause the expression to result in an irrational number.
Let's go through each option:
A. r = 3 and h = 6.54
Using the formula: V = (22/7)(3^2)(6.54)
V = (22/7)(9)(6.54)
V = (138/1)(6.54)
V = 902.52
The volume V is a rational number.
B. r = √11 and h = 6.54
Using the formula: V = (22/7)((√11)^2)(6.54)
V = (22/7)(11)(6.54)
V = (242/7)(6.54)
V = 854.34...
The volume V is a rational number.
C. r = √11 and h = √9
Using the formula: V = (22/7)((√11)^2)(√9)
V = (22/7)(11)(3)
V = (242/7)(3)
V = 726/7
The volume V is a rational number.
D. r = 3 and h = √15
Using the formula: V = (22/7)(3^2)(√15)
V = (22/7)(9)(√15)
V = (198/7)(√15)
The volume V is an irrational number because it cannot be expressed as a ratio of two integers.
Therefore, the pair of values that make V an irrational number is:
D. r = 3 and h = √15