A helpful strategy for this kind of a problem is to ask the following:
What information am I missing?
What information am I given?
Can I calculate for an unknown variable with the information I have?

The formula given is composed of the Base Area times the height of the pyramid times one-third. This means that in order to find the volume of this pyramid, the area of the square base must be calculated as well as the height of the pyramid (represented by the dashed line in the center). This will allow us to get an answer with cubed feet for units - an inkling that the volume was being calculated and not just the area. So the given equation may be expanded into: . 

Calculating the base area:
Even though pyramids have a square base on which they're commonly seen to be standing on, the base area is calculated for the triangular face. Note that all the triangles are the same.  will be used to calculate the base area.

Calculating the height of the pyramid:
The height of the pyramid is slightly more tricky than calculating B. Imagine creating a right triangle within the pyramid, which resembles a slpice from the exterior of one of the sides of the pyramid to the center, where the height would be measured. Such triangle is outlined by the dash marks in the following figure:

The hypotenuse of this triangle would be the height of the triangular face. The pyramid's height becomes to be one of the legs of the triangle - this is our unknown, and may be labeled as x. The other leg of the triangle is 5ft because the leg only extends the length of half of the base of one of the triangles (from one edge to the center). 

Because there are no angles given and two sides of the right triangle can be deduced, x (h of pyramid) can be calculated through the Pythogorean Theorem. , where x=-3.3, +3.3. However, distance cannot be negative so x=-3.3 is not a viable option and the height is left to be as 3.3ft. 

Now all information may be substituted into the given equation to solve for the final volume:

When reading word problems, there are certain clues that help interpret what is going on. The word “is” generally means “=” and the word “times” means it will be multiplied by something. Therefore, “the length of a box is 3 times the width” gives you the answer: L = 3 x W, or L = 3W.

We notice the width is “5 inches less than three times its width,” so we express W as being three times its width (3L) and 5 inches less than that is 3L minus 5. In this case, W is the dependent and L is the independent variable.

W = 3L - 5

This sounds like a geometry problem, so start by drawing a picture so that you know exactly what you are dealing with.

Because we are dealing with rectangular prisms and volume, we will need the following formula:

Or

We are solving for height, so you can begin by rearranging the equation to get  by itself:

Then, plug in our knowns (,  and )

Here is the problem worked out with a corresponding picture:

The trickiest part of this question is the wording. This problem is asking for the height of the water in the exhibit if the exhibit is three-quarters full. We can find this at least two different ways.

1) The longer way requires that we begin by finding three quarters of the total volume:

Now we go back to our volume equation, and since we are again looking for height, we want it solved for :

Becomes

2) The easier way requires that we recognize a key detail. If we take three-quarters of the volume without changing our length or width, our new height will just be three-quarters of the total height. We can solve for the total height of the exhibit by using the volume equation and rearranging it to solve for :

At this point, we can substitute in our given values and solve for :

So, the total height of the exhibit is . We can now easily solve for three-quarters of the total height:

First we must find the diagonal of the prism's base (). This can be done by using the Pythagorean Theorem with the length () and width ():

Therefore, the diagonal of the prism's base is . We can then use this again in the Pythagorean Theorem, along with the height of the prism (), to find the diagonal of the prism ():

Therefore, the length of the prism's diagonal is .

In order to solve this problem, it's helpful to visualize where the diagonal is within the prism.

In this image, the diagonal is the pink line. By noting how it relates to the blue and green lines, we can observe how the pink line is connected and creates a right triangle. This very quickly becomes a problem that employs the Pythagorean theorem. 

The goal is essentially to find the hypotenuse of this sketched-in right triangle; however, only one of the legs is given: the green line, the height of the prism. The blue line can be solved for by understanding that it is the measurement of the diagonal of a 4x4 square.

Either using trig functions or the rules for a special 45/45/90 triangle, the blue line measures out to be . 

The rules for a 45/45/90 triangle: both legs are "" and the hypotenuse is "". Keep in mind, this is is only for isosceles right triangles. 

Now that both legs are known, we can solve for the hypotenuse (diagonal).

Write the diagonal formula for a rectangular prism.

Substitute and solve for the diagonal.

Convert the dimensions into feet.

The new dimensions of rectangular prism in feet are: 

Write the formula for the diagonal of a right rectangular prism and substitute.

Find the surface area of the walls: SA_walls = 2lh + 2wh, where the height is 8 ft, the width is 10 ft, and the length is 16 ft.

This gives a total surface area of 416 ft². One gallon covers 300 ft², and each quart covers 75 ft², so we need 1 gallon and 2 quarts of paint to cover the walls.