An isosceles triangle has two 10.0-inch sides and a 2w-inch side. Find the radius of the inscribed circle of this triangle, in the cases w = 5.00, w = 6.00, and w = 8.00.

To find the radius of the inscribed circle of an isosceles triangle, we need to use the formula:

radius = (area of the triangle) / (semiperimeter of the triangle)

The semiperimeter of the triangle can be found by adding the lengths of all three sides and dividing by 2:

semiperimeter = (10.0 + 10.0 + 2w) / 2

Now, let's calculate the radius for each given value of w:

1. For w = 5.00:
semiperimeter = (10.0 + 10.0 + 2 * 5.00) / 2 = 15.0
The area of the triangle can be calculated using Heron's formula:
area = sqrt(s * (s - a) * (s - b) * (s - c)), where s is the semiperimeter, and a, b, c are the side lengths.
area = sqrt(15.0 * (15.0 - 10.0) * (15.0 - 10.0) * (15.0 - 2 * 5.00)) = 25.0
radius = area / semiperimeter = 25.0 / 15.0 = 1.67 inches

2. For w = 6.00:
semiperimeter = (10.0 + 10.0 + 2 * 6.00) / 2 = 16.0
area = sqrt(16.0 * (16.0 - 10.0) * (16.0 - 10.0) * (16.0 - 2 * 6.00)) = 24.0
radius = area / semiperimeter = 24.0 / 16.0 = 1.50 inches

3. For w = 8.00:
semiperimeter = (10.0 + 10.0 + 2 * 8.00) / 2 = 18.0
area = sqrt(18.0 * (18.0 - 10.0) * (18.0 - 10.0) * (18.0 - 2 * 8.00)) = 20.0
radius = area / semiperimeter = 20.0 / 18.0 = 1.11 inches

Therefore, the radius of the inscribed circle for w = 5.00 is 1.67 inches, for w = 6.00 is 1.50 inches, and for w = 8.00 is 1.11 inches.