the volume of an equilateral triangular frustum is 183 square root of 3 cubic inches. The upper base, 9in above the lower base, has an edge which measures 8in. Compute for the edge of the lower base.
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To compute for the edge of the lower base of the equilateral triangular frustum, we'll need to use the formula for the volume of a frustum.
The formula for the volume of a frustum is given by:
V = (1/3) * π * h * (R² + r² + R*r)
In this case, since the frustum is an equilateral triangular frustum, the top and bottom bases are equilateral triangles.
To find the edge length of the lower base, we can use the given information that the upper base is 9 inches above the lower base and that the edge length of the upper base is 8 inches.
Let's assign variables to the values given:
h = 9 inches (height of the frustum)
R = 8 inches (edge length of the upper base)
To solve for the edge length of the lower base (r), we need to rearrange the formula by substituting the given values and solving for r.
V = (1/3) * π * h * (R² + r² + R*r)
183√3 = (1/3) * π * 9 * (8² + r² + 8r)
Now, we can simplify the equation further:
183√3 = (1/3) * π * 9 * (64 + r² + 8r)
Multiply through by 3 to get rid of the fraction:
549√3 = π * 9 * (64 + r² + 8r)
Divide both sides by 9π to isolate the equation for r:
183√3/ (9π) = 64 + r² + 8r
Now, we have:
20.339 = 64 + r² + 8r
Rearranging the equation and bringing all terms to one side:
r² + 8r + 64 - 20.339 = 0
Now, we have a quadratic equation in terms of r:
r² + 8r + 43.661 = 0
Solving this quadratic equation will give us the value of r, which is the edge length of the lower base. We can use the quadratic formula to find the values of r.
r = (-b ± √(b² - 4ac)) / (2a)
Using the quadratic formula, with a = 1, b = 8, and c = 43.661, we can calculate the possible values of r. One of the values will be meaningful since we are dealing with a length measurement.
r = (-8 ± √(8² - 4 * 1 * 43.661)) / (2 * 1)
r = (-8 ± √(64 - 174.644)) / 2
r = (-8 ± √(-110.644)) / 2
Since we are dealing with a length measurement, we discard the negative value inside the square root because it is not meaningful. Square roots of negative numbers are imaginary. Therefore, we only consider the positive square root.
r = (-8 + √(-110.644)) / 2
r = (-8 + (√110.644)i) / 2
The edges of a geometric figure cannot be imaginary, so it seems that there is no real solution for the edge length of the lower base. The given information may be incorrect, or there may be a mistake in the problem statement.