(c) How many numbers between 75 and 500 are divisible by 7?
(a) If [tex]$R$[/tex] varies inversely as the square of [tex]$(3 q-2)$[/tex] and [tex]$R=4$[/tex] when [tex]$q=2$[/tex], find
Answer (1)
Find the smallest multiple of 7 greater than 75: 11 × 7 = 77 .
Find the largest multiple of 7 less than 500: 71 × 7 = 497 .
Calculate the number of multiples of 7 between 77 and 497: 71 − 11 + 1 = 61 .
Determine the relationship between R and q : R = ( 3 q − 2 ) 2 64 .
61
Explanation
Problem Analysis We are asked to find how many numbers between 75 and 500 are divisible by 7. Also, we need to find the relationship between R and q , given that R varies inversely as the square of ( 3 q − 2 ) and R = 4 when q = 2 .
Finding the Range First, let's find the number of integers between 75 and 500 that are divisible by 7. We need to find the smallest multiple of 7 that is greater than 75 and the largest multiple of 7 that is less than 500.
Smallest Multiple To find the smallest multiple of 7 greater than 75, we divide 75 by 7: 75/7 ≈ 10.71 . We round up to the next whole number, which is 11. So, the smallest multiple of 7 greater than 75 is 11 × 7 = 77 .
Largest Multiple To find the largest multiple of 7 less than 500, we divide 500 by 7: 500/7 ≈ 71.43 . We round down to the previous whole number, which is 71. So, the largest multiple of 7 less than 500 is 71 × 7 = 497 .
Counting Multiples Now we need to find how many multiples of 7 there are between 77 and 497, inclusive. The multiples of 7 are 77, 84, 91, ..., 497. We can express these as 7 × 11 , 7 × 12 , 7 × 13 , ... , 7 × 71 . So we want to find how many integers there are from 11 to 71, inclusive. This is 71 − 11 + 1 = 61 . Therefore, there are 61 numbers between 75 and 500 that are divisible by 7.
Finding the Constant of Proportionality Next, let's find the relationship between R and q . Since R varies inversely as the square of ( 3 q − 2 ) , we can write this as R = ( 3 q − 2 ) 2 k , where k is the constant of proportionality. We are given that R = 4 when q = 2 . Substituting these values into the equation, we get 4 = ( 3 ( 2 ) − 2 ) 2 k = ( 6 − 2 ) 2 k = 4 2 k = 16 k .
Relationship between R and q Solving for k , we multiply both sides by 16: k = 4 × 16 = 64 . So the relationship between R and q is R = ( 3 q − 2 ) 2 64 .
Final Answer Therefore, the number of integers between 75 and 500 that are divisible by 7 is 61, and the relationship between R and q is R = ( 3 q − 2 ) 2 64 .
Examples
Understanding divisibility is crucial in many real-world scenarios, such as evenly distributing resources or scheduling events. For instance, if you have 500 items to distribute among groups and want to ensure each group receives a multiple of 7 items to maintain fairness, knowing how many multiples of 7 exist below 500 helps in planning. Similarly, inverse variation is used in physics to describe relationships like the intensity of light decreasing with the square of the distance from the source. This concept is vital in designing lighting systems and understanding wave propagation.
