suppose r varies directly as the square of m and inversely as s. if r=20 when m=15 and s=9, find r when m=30 amd s=9

Question

suppose r varies directly as the square of m and inversely as s. if r=20 when m=15 and s=9, find r when m=30 amd s=9

Answer 1

r varies directly as the m means that r = km^2 where k is some constant number.

and r varies inversely as s means r = k/s where k is also some constant.

Putting these together tells us that r = km^2/s

 

Let's say we have NO idea how to do this problem but we found this equation and just plug in the numbers they gave us to find r:

 

we get, r = k*(30)^2/9

We now have a problem! We don't know k or r. But we can fix that by using the rest of the information that they gave us.

We know that r = 20 when m = 15 and s = 9

 

This gives us

20 = k(15)^2/9

Now we only don't know k! We can find that so let's solve for k:

 

k = (20 * 9)/225 which is 0.8

 

Now let's just go back to the original "problematic" equation since we now know k:

 

r = .8 * (30^2)/ 9 = 0.8 * 900 / 9 = 80, so we now found r to be 80.

 

Now let's just do a "does this make sense" analysis of the problem because I feel that is  very important to the learning process. So we notice that s stayed the same in both problems, so that really shouldn't change anything right? And we can also see that we doubled m. Since r varies as the square of m, this means that r at 30 should be 2^2 times the r at 15. 80 IS in fact 4 times 20 and therefore it makes perfect sense :)