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 = 22/7 r2h
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 which pair of values makes V an irrational number, we need to find the volume V for each pair and determine if it can be simplified to a rational number.
A. r=3 and h=6.54:
V = (22/7)(3^2)(6.54) = (22/7)(9)(6.54) = (198/7)(6.54) = 1291.714...
This value is not an irrational number.
B. r=√11 and h=6.54:
V = (22/7)((√11)^2)(6.54) = (22/7)(11)(6.54) = (242/7)(6.54) = 1315.428...
This value is not an irrational number.
C. r=√11 and h=√9:
V = (22/7)((√11)^2)(√9) = (22/7)(11)(3) = 94.285...
This value is not an irrational number.
D. r=3 and h=√15:
V = (22/7)(3^2)(√15) = (22/7)(9)(√15) = (198/7)(√15) = 535.714...
This value is not an irrational number.
None of the given pairs of values make V an irrational number.
To determine which pair of values makes the volume V an irrational number, we need to substitute the given values into the formula and check if the result is an irrational number.
Let's evaluate the formula using each of the given value pairs:
A. r = 3 and h = 6.54:
V = (22/7) * (3^2) * 6.54 = (22/7) * 9 * 6.54 = 205.886...
B. r = √11 and h = 6.54:
V = (22/7) * (√11^2) * 6.54 = (22/7) * 11 * 6.54 = 429.942...
C. r = √11 and h = √9:
V = (22/7) * (√11^2) * (√9) = (22/7) * 11 * 3 = 94.285...
D. r = 3 and h = √15:
V = (22/7) * (3^2) * (√15) = (22/7) * 9 * 3.873 = 97.285...
Based on the evaluations above, we can see that none of the calculated values are irrational numbers. An irrational number is one that cannot be expressed as a fraction and has an infinite number of non-repeating decimal places. Since all the calculations result in decimal values, none of the options make the volume V an irrational number.
Therefore, the answer is none of the given pairs satisfy the condition of making V an irrational number.
To determine which pair of values makes the volume V an irrational number, we need to substitute the values of r and h into the formula and check if the resulting volume is irrational.
Let's consider each option:
A. r = 3 and h = 6.54
Substituting these values into the formula:
V = (22/7)(3^2)(6.54)
V = (22/7)(9)(6.54)
V = (22)(9)(6.54)/7
V = (22)(59.04)/7
V = 1296.64
Since 1296.64 is a rational number (can be expressed as a fraction), option A does not make V an irrational number.
B. r = √11 and h = 6.54
Substituting these values into the formula:
V = (22/7)(√11^2)(6.54)
V = (22/7)(11)(6.54)
V = (22)(11)(6.54)/7
V = (22)(72.94)/7
V = 2371.42857143
Since 2371.42857143 is a rational number, option B does not make V an irrational number.
C. r = √11 and h = √9
Substituting these values into the formula:
V = (22/7)(√11^2)(√9)
V = (22/7)(11)(3)
V = (22)(11)(3)/7
V = (22)(33)/7
V = 102
Since 102 is a rational number, option C does not make V an irrational number.
D. r = 3 and h = √15
Substituting these values into the formula:
V = (22/7)(3^2)(√15)
V = (22/7)(9)(√15)
V = (22)(9)(√15)/7
Since √15 is an irrational number, the volume V in this case will also be irrational. Therefore, option D makes V the volume in cubic meters an irrational number.
Therefore, the correct answer is option D. r = 3 and h = √15.