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Problem 854

# The base of a right prism, as shown in the figure, is an equilateral triangle, each of whose sides measures 4 units. The altitude of the prism measures 5 units. Find the volume of the prism. [Leave answer in radical form.]

Expert verified
The volume of the right prism with an equilateral triangular base of side 4 units and altitude of 5 units is $$20\sqrt{3}$$ cubic units.
See the step by step solution

## Step 1: Find the area of the equilateral triangle base

To find the area of an equilateral triangle with side length 'a', we need to use the formula: $Area = \frac{\sqrt{3}}{4}a^2$ In our case, the side length (a) is 4 units. Now, we will calculate the area of the triangle using the given formula: $Area = \frac{\sqrt{3}}{4}(4^2)$

## Step 2: Simplify the expression of the area

After substituting the value of a, now we simplify the expression: $Area = \frac{\sqrt{3}}{4}(16)$ $Area = 4\sqrt{3}$

## Step 3: Find the volume of the prism

Now we have the area of the base of the prism (4√3) and the altitude of the prism (5). The formula for finding the volume of a prism is: $Volume = (Area \space of \space base) \times h$ Here, h stands for the altitude (height) of the prism. Substituting the area of the base and altitude in the equation, we get: $Volume = (4\sqrt{3}) \times 5$

## Step 4: Simplify the expression and write the final answer in radical form

Finally, we simplify the expression to get the volume of the prism: $Volume = 20\sqrt{3}$ So, the volume of the right prism with an equilateral triangular base of side 4 units and altitude of 5 units is $$20\sqrt{3}$$ cubic units.

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