Question 128·Medium·Math

You need the area of an equilateral triangle with side length 6. You can either use the special formula for the area of an equilateral triangle, or find the triangle's height using a 30-60-90 triangle and then use $A = \tfrac12 bh$.

Once you have the area of the equilateral triangle, multiply that area by the prism's height of 10 to find the volume.

In Desmos, type the expression for the area of the equilateral triangle base: (sqrt(3)/4)*6^2. Note the value Desmos gives for this area.

Now multiply the base area by the prism height: type (sqrt(3)/4)*6^2*10. This expression represents the volume of the prism; record the numerical value shown by Desmos.

In separate lines, type each answer choice as an expression (for example, 90*sqrt(3), 108*sqrt(3), 180, 360). Compare their numerical values to the volume you computed in Step 2. The choice whose value matches the volume is the correct answer.

For any right prism,

Here:

• The base is an equilateral triangle with side length 6 cm.
• The height of the prism is 10 cm.

So we first need the area of the equilateral-triangle base.

For an equilateral triangle with side length $s$, the area formula is

Substitute $s = 6$:

So the area of the base is $9\sqrt{3}$ square centimeters.

Now use $V = (\text{area of base}) \times (\text{height})$ with base area $9\sqrt{3}$ and height $10$:

So the volume of the prism is $90\sqrt{3}$ cubic centimeters.