How to use the direct variation formula

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ACT Math › How to use the direct variation formula

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In an experimental setting, the pressure and the temperature of gas in a container are directly proportional. In the first experiment, the pressure of the gas was and the temperature of the gas was , where is a constant value not equal to . If a second experiment found a temperature of , which of the following represents the pressure of the gas?

$\frac{24x+120}{x}$

$\frac{24x^2+15}{x^2}$

$\frac{12x^2+10}{x^2}$

$\frac{23x^2+150}{2x}$

It cannot be determined.

Explanation

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

$\frac{p_0}{t_0}=\frac{p_1}{t_1}$

For our data, this would be:

$\frac{3x+15}{3x^2} = \frac{p}{24x}$

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

$\frac{3(x+5)}{3x^2} = \frac{p}{24x}$

Cancel the s:

$\frac{x+5}{x^2} = \frac{p}{24x}$

Next, multiply by :

$p = \frac{24x(x+5)}{x^2}$

Since does not equal , cancel the s:

$p = \frac{24(x+5)}{x}$

Simplify:

$p = \frac{24x+120}{x}$

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?

Explanation

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:

$\frac{x}{y} = \frac{a}{b}$

Thus, for our information, we know:

$\frac{40}{8} = \frac{a}{b}$

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

In a given solution, the proportion of water to apple juice is directly proportional. If the first batch of the solution contained gallons of apple juice and gallons of water, how many gallons of apple juice will be needed for a solution containing gallons of apple juice?

Explanation

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

$\frac{w_0}{a_0} = \frac{w_1}{a_1}$

For our data, this would be:

$\frac{140}{30} = \frac{w_1}{75}$

This is first simplified as being:

$\frac{14}{3} = \frac{w_1}{75}$

Next, multiply by on both sides to solve for :

$\frac{1050}{3}=w_1$

Finally:

The length, in inches, of a box is $\textup{5 inches less than twice its width}$ . Which of the following gives the length, inches, in terms of the width, inches, of the box?

$l=\left ( \frac{1}{2} \right )w-5$

$l=\left ( \frac{1}{2} \right )w+5$

Explanation

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 $\textup{5 less}$ 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.

The diameter of a specific brand of candy wrapper is longer than half the volume of the candy itself. Find the expression for the diameter, , in terms of the volume, .

$d=\frac{4}{3}v^2$

$d=\frac{1}{2}v-6$

$d=\frac{1}{2}v+6$

Explanation

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:

$d=\frac{1}{2}v+6$

The price of silver varies directly as the square of the mass. If 3.6 g of silver is worth $64.80, what is the value of 7.5 g of silver?

$135.00

$301.75

$178.50

$215.25

$281.25

Explanation

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.

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