This is a direct variation problem of the form y = kx²  The first set of data 3.6 g and $64.80 is used to calculate the proportionality constant, k.  So 64.80 = k(3.6)² and solving the equation gives k = 5.

Now we move to the new data, 7.5 g and we get y = 5(7.5)² to yield an answer of $218.25.

$135.00 is the answer obtained if using proportions.  This is an error because it does not take into consideration the squared elements of the problem.

The question asks for an equation that can relate  and  to each other, based on the information given. We are told that half the volume +  determines the total diameter.

This gives us:

An instrument reads two values,  and  daily. The values directly vary with respect to each other. If on Monday the value of  was  and  was , which of the following could be the values for  and  on Wednesday?

Direct variation means that any pairing of the related values will always have the same ratio, thus we know that for any other values  and , those values will be equal according to the following equation:

Thus, for our information, we know:

This means that the new values of  and , when divided must be equal to . Therefore, the only possible answer is 

For direct variation of related variables, we know that the following equation holds:

For our data, this would be:

To start simplifying and solving this, first factor the top of the left fraction:

Cancel the s:

Next, multiply by :

Since  does not equal , cancel the s:

Simplify:

This problem is easily solved by setting up a ratio. For directly proportional amounts, we know:

For our data, this would be:

This is first simplified as being:

Next, multiply by  on both sides to solve for :

Finally:

Since the question asks for the length in terms of width, I like to start by setting the problem up as , then add the conditions off the problem. I start at the first condition with  than the width and go ahead and add a  after , then it say  less than TWICE the width, so I add my  in front of the . This gives you ,  essentially you're working the problem piece by piece.

1. Use the given values of  and  to solve for :

2. Solve for  when  in the above equation:

To solve this algebraic equation, subtract  from both sides, and then subtract  from both sides.

We end up with the equation , for which the solution is:

Recall that inverse variation means that when one variable increases, the other decreases. This gives you the following equation:

Now, for our data, we know:

You merely have to solve for :

Divide by :

Recall that inverse variation means that when one variable increases, the other decreases. This gives you the following equation:

For our data, this means:

You merely need to solve for :