Answer:
Explanation:
We can use the formula for the volume of a pyramid, which is:
V = (1/3) * B * h
where V is the volume, B is the area of the base, and h is the height.
In this case, we have a triangular pyramid, so the base is a triangle. Let's call the length of the base x, and the height of the pyramid h. Then, the area of the base is:
B = (1/2) * x * h
Substituting into the formula for the volume, we get:
256 = (1/3) * (1/2) * x * h * h
Simplifying and solving for h, we get:
256 = (1/6) * x * h^2
h^2 = (256 * 6) / x
h = sqrt((256 * 6) / x)
Now, let's use the given information that the three sides of the base have lengths x, 2x, and 3x, to find the area of the base:
B = (1/2) * x * (2x + 3x) / 2
B = (5/4) * x^2
Substituting this and the expression for h into the formula for the volume, we get:
256 = (1/3) * (5/4) * x^2 * sqrt((256 * 6) / x)^2
Simplifying, we get:
256 = (5/4) * x^2 * sqrt(1536 / x)
256 = (5/4) * x^2 * (sqrt(1536) / sqrt(x))
256 = (5/4) * x^2 * (12 / sqrt(x))
256 = 15 * x * sqrt(x)
Squaring both sides, we get:
65536 = 225 * x^3
Dividing both sides by 225, we get:
x^3 = 65536 / 225
Taking the cube root of both sides, we get:
x = (65536 / 225)^(1/3)
x ≈ 6.4
Therefore, the value of x is approximately 6.4 units.