# Dakota's travel toothbrush case, shaped like an equilateral triangular prism, has perpendicular height of 2.3 in. and depth of 8.4 in. How much plastic is the case made out of, in sq. in.?

Oct 5, 2017

About $28.42$ square inches.

#### Explanation:

Assuming this includes plastic to cover the three long sides as well as the two triangular ends, Let's look at the ends first:
The perpendicular height of the triangle is an altitude drawn from the top vertex, which gives you two 30-60-90 triangles on either end.
On questions like this, always draw a picture first!!

You can use the law of sines to set it up like this: $\frac{\sin {60}^{o}}{2.3} = \frac{\sin {90}^{o}}{x}$, with $x$ representing the length of a side of the triangle.

Or you can use your 30-60-90 rules, which say that the long leg (the perpendicular height) is $\frac{\sqrt{3}}{2} \cdot$the hypotenuse and solve for $x$ that way. $2.3 \sqrt{3} = 2 x \implies \frac{2.3 \sqrt{3}}{2} = x \approx 2.66$.
Either method gives you a value where $x = \frac{2.3}{\sqrt{3}} \approx 2.66$.
To find the area of the triangles, use the formula Area=1/2 base*height. The area of each triangle ends up being $2.3 \frac{x}{2} \approx 3.054$.

For the long sides, you just have three congruent rectangles. Taking that value of $x$ from before, the rectangles have short sides$= x$ in. and long sides$= 8.4$ in. To find the area, length$\cdot$width, you have $8.4 x \approx 22.31$ in.

Three of these plus the two triangles on the end, the surface area is about $28.42$ square inches.