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

First we must calculate the surface area of the cube. We know that there are six surfaces and each surface has the same area:

\(\displaystyle Area=6(2^2)=6\times 4=24\hspace{1 mm}feet^2\)

Now we will determine the amount of paint needed

\(\displaystyle 24\hspace{1 mm}feet^2\times \frac{25\hspace{1 mm}mL}{1\hspace{1 mm}foot^2}=600\hspace{1 mm}mL\)

The formula for the surface area of a triangular prism is:

\(\displaystyle SA = 2 (base) + lateral\ area\)

\(\displaystyle SA = 2 \cdot \left(\frac{1}{2}\right)(l)(w) + (l+w+y)(h)\)

Where \(\displaystyle l\) is the length of the triangle, \(\displaystyle w\) is the width of the triangle, \(\displaystyle y\) is the hypotenuse of the triangle, and \(\displaystyle h\) is the height of the prism

Use the formula for a \(\displaystyle 45-45-90\) triangle to solve for the length of the hypotenuse:

\(\displaystyle s-s-s\sqrt{2}\)

\(\displaystyle 6-6-6\sqrt{2}\)

Plugging in our values, we get:

\(\displaystyle SA = (6m)(6m) + (6m+6m+6\sqrt{2}m)(9m)\)

\(\displaystyle SA = (6m)(6m) + (12m + 6\sqrt{2}m)(9m)\)

\(\displaystyle SA = 36m^2 + 108m^2 + 54\sqrt{2}m^2 = 144 + 54\sqrt{2} m^2\)

The formula for the surface area of a triangular prism is:

\(\displaystyle SA = 2(base)+(perimeter)(height)\)

\(\displaystyle SA = 2\left(\frac{1}{2}\right)(l)(w) + (l+w+y)(h)\)

\(\displaystyle SA = (l)(w) + (l+w+y)(h)\)

Where \(\displaystyle l\) is the length of the base, \(\displaystyle w\) is the width of the base, \(\displaystyle y\) is the hypotenuse of the base, and \(\displaystyle h\) is the height of the prism

Use the formula for a \(\displaystyle 45-45-90\) triangle to find the length of the hypotenuse:

\(\displaystyle a-a-a\sqrt{2}\)

\(\displaystyle 8m-8m-8\sqrt{2}m\)

Plugging in our values, we get:

\(\displaystyle SA = (8m)(8m) + (8m+8m+8\sqrt{2}m)(12m)\)

\(\displaystyle SA = 256m^2 + 96\sqrt{2}m^2\)

The formula for the surface area of an equilateral, triangular prism is:

\(\displaystyle SA = 2(base)+(perimeter)(height)\)

\(\displaystyle SA = 2\left(\frac{s^2\sqrt{3}}{4}\right)+(3s)(h)\)

Where \(\displaystyle s\) is the length of the triangle side and \(\displaystyle h\) is the length of the height.

Plugging in our values, we get:

\(\displaystyle SA = 2\left(\frac{(6m)^2\sqrt{3}}{4}\right)+(3)(6m)(12m)\)

\(\displaystyle SA = 216 + 18\sqrt{3}m^2\)

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.

The box will have six total faces: an identical "top and bottom," and identical "left and right," and an identical "front and back." The total surface area will be the sum of these faces.

Since the six faces consider of three sets of pairs, we can set up the equation as:

\(\displaystyle SA=2(\text{top})+2(\text{left})+2(\text{front})\)

Each of these faces will correspond to one pair of dimensions. Multiply the pair to get the area of the face.

\(\displaystyle SA=2(lw)+2(wh)+2(lh)\)

Substitute the values from the question to solve.

\(\displaystyle SA=2(5*5)+2(5*4)+2(5*4)\)

\(\displaystyle SA=50+40+40\)

\(\displaystyle SA=130\ in^2\)

The formula for the surface area of a rectangular prism is given by:

SA = 2LW + 2WH + 2HL

SA = 2(12 * 8) + 2(8 * 6) + 2(6 * 12)

SA = 2(96) + 2(48) + 2(72)

SA = 192 + 96 + 144

SA = 432 in²

216 in²  is the wrong answer because it is off by a factor of 2

576 in³ is actually the volume, V = L * W * H 

The surface area of the prism can be broken into three rectangular sides and two equilateral triangular bases.

The area of the sides is given by:  \(\displaystyle A=lw=12\cdot 6=72\ in^{2}\), so for all three sides we get \(\displaystyle \dpi{100} 216\ in^{2}\).

The equilateral triangle is also an equiangular triangle by definition, so the base has congruent sides of 6 in and three angles of 60 degrees.  We use a special right traingle to figure out the height of the triangle: 30 - 60 - 90.  The height is the side opposite the 60 degree angle, so it becomes \(\displaystyle 3\sqrt{3}\) or 5.196. 

The area for a triangle is given by \(\displaystyle A=\frac{1}{2}bh\) and since we need two of them we get \(\displaystyle A=bh=6\cdot 3\sqrt{3}=31.176\ in^{2}\).

Therefore the total surface area is \(\displaystyle 216\ in^{2}+31.176\ in^{2}=247.176\ in^{2}\).