The general formula for direct proportionality is 

\(\displaystyle M=kh\)

where \(\displaystyle M\) is how much money you earned, \(\displaystyle k\) is the proportionality constant, and \(\displaystyle h\) is the number of hours worked.

Before we can figure out how many hours you need to work to earn $48, we need to find the value of \(\displaystyle k\). It is given that you earned $32 by working 4 hours. Plug these values into the formula

\(\displaystyle 32=4k\)

Solve for \(\displaystyle k\) by dividing both sides by 4.

\(\displaystyle \frac{32}{4}=\frac{4k}{4}\)

\(\displaystyle 8=k\)

So \(\displaystyle k=8\). We can use this to find out the hours you need to work to earn $48. With \(\displaystyle k=8\), we have

\(\displaystyle M=8h\)

Plug in $48.

\(\displaystyle 48=8h\)

Divide both sides by 8

\(\displaystyle \frac{48}{8}=\frac{8h}{8}\)

\(\displaystyle 6=h\)

So you will need to work 6 hours to earn $48.

We must set up a proportion. Since x varies directly with y, when y is multiplied by 2, x is also multiplied by 2. 26 times 2 is 52.

Direct variation: \(\displaystyle \frac{x_1}{x_2}=\frac{y_1}{y_2}\)

\(\displaystyle \frac{26}{x_2}=\frac{100}{200}\)

\(\displaystyle \frac{26}{x_2}=\frac{1}{2}\)

\(\displaystyle x_2=2(26)=52\)

Let \(\displaystyle y\) be the number of pizzas and \(\displaystyle x\) be the number of people attending. Because \(\displaystyle y\) is directly proportional to \(\displaystyle x\), we have

\(\displaystyle y=kx\)

and from the problem

\(\displaystyle 6=12k\).

Therefore, solving for \(\displaystyle k\) we get:

\(\displaystyle k=0.5,\,\,\, y=0.5x\)

Now solving for \(\displaystyle x=18\),

\(\displaystyle y=0.5(18)=9\)

Thus, for 18 people 9 pizzas are needed.

Direct proportionality results when two quantities increase or decrease at the same time. In the formula for resistance, \(\displaystyle R=\frac{\rho L}{A}\), as \(\displaystyle L\) increases so too does \(\displaystyle R\). The same is true for \(\displaystyle \rho\).  It helps to think of these variables in terms of numbers. When the numerator of this formula is increased and the denominator remains the same, the overall quotient would be larger. The opposite is true if the numerator is decreased and the denominator remains the same (quotient would be smaller).

Direct proportionality means that as one increases so does the other. The equation that is indicative of this statement is the following:

\(\displaystyle \small y=kx\)

Direct proportionality means you should use the equation \(\displaystyle y=kx\)

The question says that the amount of money earned by the business is directly proportional to the number of driveways shoveled so that means y must be the amount of money earned ($3000) and x must be the amount of driveways shoveled (600). Plugging these values into the equation you should get

\(\displaystyle 3000=600k\)

Solve for k by doing the opposite of what is being done to k on both sides of the equation. Since k is being multiplied by 600 divide both sides by 600

\(\displaystyle \frac{3000}{600}=k\)

\(\displaystyle k=5\)

Each driveway earned the snow shoveling business $5

Breaking down the question one small piece at a time:

"The force from gravity (\(\displaystyle g\)) is directly proportional" means that the gravity term is going to be on one side of the proportional symbol, and everything else is going to be on the other side.

"to the inverse...of the distance" means that the distance will be in the denominator.  As distance gets larger, force will get smaller.

"of the square of the distance" means that the distance will be squared.

Putting it all together gives:

\(\displaystyle g\propto \frac{1}{d^{2}}\)

The variation equation can be written as below. Direct variation will put \(\displaystyle y\) in the numerator, while inverse variation will put \(\displaystyle \sqrt{z}\) in the denominator. \(\displaystyle k\) is the constant that defines the variation.

\(\displaystyle x=k* \frac{y}{\sqrt{z}}\)

To find constant of variation, \(\displaystyle k\), substitute the values from the first scenario given in the question.

\(\displaystyle 10=k* \frac{6}{\sqrt{64}}=k* \frac{6}{8}=k* \frac{3}{4}\)

\(\displaystyle k=10\div\frac{3}{4}}= \frac{40}{3}\)

We can plug this value into our variation equation.

\(\displaystyle x=\frac{40}{3}\cdot \frac{y}{\sqrt{z}}\)

Now we can solve for \(\displaystyle x\) given the values in the second scenario of the question.

\(\displaystyle x=\frac{40}{3}* \frac{9}{\sqrt{144}}=\frac{40}{3}* \frac{9}{12}=10\)

The variation equation is \(\displaystyle A = \frac{K}{\sqrt{x}}\) for some constant of variation \(\displaystyle K\).

Substitute the numbers from the first scenario to find \(\displaystyle K\):

\(\displaystyle 45= \frac{K}{\sqrt{15}}\) 

\(\displaystyle K = 45 \sqrt{15}\)

The equation is now \(\displaystyle A = \frac{45 \sqrt{15}}{\sqrt{x}}\).

If \(\displaystyle x = 30\), then

\(\displaystyle A = \frac{45 \sqrt{15}}{\sqrt{30}} \approx 31.8\)

In order to find the value of \(\displaystyle y\) when \(\displaystyle x=100\), first determine the variation equation based on the information provided:

\(\displaystyle y=\frac{K}{3\sqrt{x}}\), for some constant of variation \(\displaystyle K\).

Insert the \(\displaystyle x\) and \(\displaystyle y\) values from the first variance to find the value of \(\displaystyle K\):

\(\displaystyle 2=\frac{K}{3\sqrt{81}}\)

\(\displaystyle 2=\frac{K}{(3\times 9)}\)

\(\displaystyle 2=\frac{K}{27}\)

\(\displaystyle 54=K\)

Now that we know \(\displaystyle K=54\), the variation equation becomes:

\(\displaystyle y=\frac{54}{3\sqrt{x}}\)

or  

\(\displaystyle y=\frac{18}{\sqrt{x}}\).

Therefore, when \(\displaystyle x=100\):

\(\displaystyle y=\frac{18}{\sqrt{100}}\)

\(\displaystyle y=\frac{18}{10}\)

\(\displaystyle y=1.8\)