Final answer:
To find the side length of the equilateral triangle cross-section of a prism with given volume and length, we use the formula for the volume of a prism and the area formula for an equilateral triangle to set up an equation and solve for the side length, yielding approximately 5.77 cm.
Step-by-step explanation:
The question asks us to find the length of a side of an equilateral triangle which forms the cross-sectional area of a prism with a volume of 100 cm3 and a length of 8 cm. The formula for the volume of a prism is V = Ah × h, where Ah is the area of the cross-section and h is the height (or the length of the prism). For an equilateral triangle, the area Ah can be calculated using the formula Ah = (sqrt(3)/4) × a2, where a is the length of a side of the triangle.
To find the side length a, we start by substituting the known values into the volume formula to solve for Ah:
100 cm3 = Ah × 8 cm.
Then we find Ah:
Ah = 100 cm3 / 8 cm = 12.5 cm2.
Next, we use the area formula for an equilateral triangle to solve for a:
12.5 cm2 = (sqrt(3)/4) × a2
After rearranging and taking the square root:
a = sqrt((4 × 12.5 cm2) / sqrt(3))
a ≈ 5.77 cm