Solve each problem. Suppose r varies directly as the square of m,...

Question

Solve each problem. Suppose r varies directly as the square of m, and inversely as s. If r=12 when m=6 and s=4, find r when m=6 and s=20.

Accepted Answer

Hey, everyone. Welcome back here. We're asked to solve the following variation problem. Let P vary directly as the square of Q. And inversely as R, if P is equal to 18, where Q is equal to three and R is equal to five, find P where Q is equal to eight and R is equal to 20. We're given four answer choices. Option A 32 option B option C 500 option D four. Now this is a short problem, but we're given a lot of information in this short little problem. So let's start with the first part. We're told that P varies directly as the square of Q. OK. So let's start with just that if P varies directly with something that means that it is proportional too. So we have that P is proportional to the square of Q. So P is proportional to Q squared. All right. So that's the first part. And then we have P is varies directly or sorry, varies inversely as R OK. So P varies inversely as R if we have inverse variation, that means that it's inversely proportional. And so P is proportional to one divided by R OK. So putting those two together, we have that P is proportional to Q squared divided by R OK, varies directly with Q squared and inversely with bar. Now we've written this as a proportionality. What we wanna do is convert that into an equation. In order to do that, we're gonna replace our proportional symbol with an equal sign. In order to do that, we need to introduce a constant of variation. So we can write that P is equal to some constant K multiplied by Q squared divided by R. Now, we don't know what this consonant K is yet. So if we wanna find P for a particular Q and R value, we first need to know K and this is where the other piece of information we're given. And the problem comes into play, we're told that we have P equals 18. OK? When Q is equal to three and R is equal to five. So that gives us A P A Q and an R value, we substitute that into our equation. The only unknown we have is K. So we can solve for K. And then we'll be able to use that to find the P value we're looking for. All right. So we have 18 is equal to K multiplied by Q squared multiplied by three squared divided by R which is five, this gives us 18. It's equal to nine K divided by five. We can multiply the five up to the 18. And then we're gonna divide both sides by nine to isolate for K, we get that K is equal to 18, multiplied by five, divided by nine. Simplifying this, we know that 18, divided by nine is two. So we essentially have two multiplied by five. And we get that K is equal to 10. So we have the value of this constant K. Now we know that K is equal to 10. So when we write our equation, we can write that P is equal to 10 multiplied by Q squared divided by R. And that's the exact same equation we had before. But now we know the value of K. OK. Now the question is asking us to find P and it's asking us to find P for Q value of eight and an R value of 20. So now we have P is equal to 10, that K value we found multiplied by eight squared divided by 20. This gives us that P is equal to 10, multiplied by 64 divided by 10, divided by 20 gives us from one half, multiplied by 64 gives us 32. And so we have a P value equal to 32. OK. And that's that P value we were looking for. So we started with this variation, we were able to write it in terms of proportionality, which we converted into an equation using a proportionality constant K. We used the first set of information given to solve for the value of K which allowed us to then solve for the P value we were looking for. And we found that that P value was 32 which corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

Solve each problem. Suppose r varies directly as the square of m, and inversely as s. If r=12 when m=6 and s=4, find r when m=6 and s=20.

Key Concepts

Direct Variation

Inverse Variation

Combining Direct and Inverse Variation

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