Answer:

I. Equivalent equation form for each statement:

1. The amount of money you earn is directly proportional to the number of hours you work.

Equation: Earnings = k * Hours

2. The weight of an object on the Moon varies directly with its weight on Earth.

Equation: Weight on Moon = k * Weight on Earth

3. The volume of a gas is directly proportional to its temperature in Kelvin.

Equation: Volume = k * Temperature

4. The number of people served varies directly with the amount of ground meat used to make burgers.

Equation: Number of people served = k * Amount of ground meat

5. The amount of a purchase varies directly with the number of pounds of peaches.

Equation: Purchase amount = k * Number of pounds of peaches

6. The area of a square varies directly as the square of its side.

Equation: Area = k * Side^2

7. The surface area of a square surface (A) is directly proportional to the square of either side (x).

Equation: A = k * x^2

8. The length of the buildings' shadows (s) varies directly as their height (h).

Equation: s = k * h

II. Problem Solving:

1. If y varies directly as x, and x = 12 when y = 9, what is the equation that describes this direct variation?

Using the given values, we can find the value of k:

9 = k * 12

k = 9/12

k = 3/4

Therefore, the equation that describes this direct variation is y = (3/4) * x.

2. A wooden box is made which is directly proportional to the number of wooden blocks. 120 wooden blocks are needed to make 30 boxes. How many wooden blocks are needed to prepare a box?

Using the given values, we can find the value of k:

30 = k * 120

k = 30/120

k = 1/4

Therefore, 1 wooden block is needed to prepare a box.

3. Suppose a varies directly as b and a = 30 when b = 6. What is the value of a when b = 100?

Using the given values, we can find the value of k:

30 = k * 6

k = 30/6

k = 5

Now, we can find the value of a when b = 100:

a = k * b

a = 5 * 100

a = 500

Therefore, when b = 100, the value of a is 500.