Question

An equilateral triangle has an area of (64 √3) square centimeters. If each side of the triangle is decreased by 4 cm, by how many square centimeters is the area decreased?

A. (48 √3)
B. (64 √3)
C. (16 √3)
D. (32 √3)

Answer 1

To determine the decrease in area when each side of an equilateral triangle is decreased by 4 cm, calculate the original side length, find the new side length after the decrease, calculate the area of the smaller triangle, and subtract this from the original area.

Step-by-step explanation:

The question asks how much the area of an equilateral triangle decreases if each side is shortened by 4 cm. First, we use the formula for the area of an equilateral triangle, which is A = (\sqrt{3}/4) × a^2, where a is the length of a side. Given that the original area is (64 \sqrt{3}) square centimeters, we solve for a to find the original side length. After calculating the new side length which is 4 cm less, we find the area of the new, smaller triangle and subtract it from the original area to find the decrease in area.

Step-by-Step Solution:

1. Calculate the original side length using the formula: a = \sqrt{(4A)/\sqrt{3}}.
2. Deduct 4 cm from the original side length to get the new side length: a' = a - 4.
3. Calculate the new area using the side length a' with the formula for the area of an equilateral triangle.
4. Subtract the new area from the original area to find the decrease in area.

The answer will correspond to one of the multiple-choice options provided (A, B, C, or D).