+17 At the office, mice vary inversely with cats, that is, mice = k/cats, for some value of k. When there are 3r-19 cats, there are 2r+1 mice, and when there are 6r-24 mice, there are r-3 cats. Find k.
+769 Understanding the Problem
We're given an inverse relationship between mice and cats:
Mice = k/cats
We have two sets of data:
When cats = 3r - 19, mice = 2r + 1
When mice = 6r - 24, cats = r - 3
We need to find the value of k.
Setting up the Equations
Using the given information, we can create two equations:
2r + 1 = k / (3r - 19)
6r - 24 = k / (r - 3)
Solving for k
We can solve either equation for k and then substitute it into the other equation. Let's solve the first equation for k:
k = (2r + 1)(3r - 19)
Now, substitute this expression for k in the second equation:
6r - 24 = [(2r + 1)(3r - 19)] / (r - 3)
Simplify the equation:
(6r - 24)(r - 3) = (2r + 1)(3r - 19)
6r^2 - 42r + 72 = 6r^2 - 35r - 19
7r = 91
r = 13
Now, substitute r = 13 back into either of the original equations to find k. Let's use the first equation:
k = (213 + 1)(313 - 19)
k = (27)(20)
k = 540
Therefore, the value of k is 540.
