Francisco and Valerie each calculated the volume of an equilateral triangular prism with an apothem of 4 units and a height of 5 units.
We are asked to find if either of them is correct. First we will solve it correctly. Then we will analyze their solutions.
Correct Solution
We will use the formula for the volume of a prism, V=Bh. The variable B is the area of the base, and h is the height of the prism. This tells us that h=5. We will find B. Let's analyze the base.
Notice that the height of the equilateral triangle is also its median. From the
Centroid Theorem, the centroid C is two-thirds of the distance from each vertex to the midpoint of the opposite side. This tells us that BC is two times larger than AC, which is an apothem of a length of 4 units.
BC=2* AC=2* 4=8
Now we can find the height, h=AB.
The side length of the equilateral triangle is s=8sqrt(3) units. Now, let's use the formula for the area of a regular polygon.
B=1/2aP
The variable P is the perimeter of a polygon, and a is its apothem. In our case, a=4 and P=3* 8sqrt(3)=24sqrt(3). Let's find B.
Therefore, the volume of the given prism is 240sqrt(3) cubic units.
Francisco's Solution
As we can see, Francisco's solution is almost the same as ours. Therefore, he is correct.
Valerie's Solution
Since Valerie gets an incorrect answer, she is not correct. Let's find the mistakes that she made.
She used a formula B= sqrt(3)2 s^2 for the area of a equilateral triangle. However, this formula looks like B= sqrt(3)4s^2, where s is the side length of the triangle.
She substituted 4sqrt(3) for s into the formula, but the side length of the triangle is 8sqrt(3), not 4sqrt(3).
