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.
Then Write an expression for the inscribed radius r in terms of the variable w , then find the value of w, to the nearest hundredth, that gives the maximum value of r.
To find the radius of the inscribed circle of an isosceles triangle, we can use the formula:
r = (A / s)
where A is the area of the triangle and s is the semiperimeter of the triangle.
To find the area of the triangle, we can use Heron's formula:
A = sqrt(s * (s - a) * (s - b) * (s - c))
where a, b, and c are the sides of the triangle.
In this case, we have an isosceles triangle with two sides of 10.0 inches and a side of 2w inches.
Case 1: w = 5.00
In this case, the two sides of the triangle are 10.0 inches, and the remaining side is 2 * 5.00 = 10.00 inches.
The semiperimeter can be calculated as:
s = (a + b + c) / 2 = (10.0 + 10.0 + 10.00) / 2 = 15.00 inches
The area of the triangle can be calculated as:
A = sqrt(s * (s - a) * (s - b) * (s - c))
= sqrt(15.00 * (15.00 - 10.0) * (15.00 - 10.00) * (15.00 - 10.00))
= sqrt(15.00 * 5.00 * 5.00 * 5.00)
= sqrt(3750.00)
≈ 61.23 square inches
Now, we can calculate the radius of the inscribed circle using the formula:
r = A / s = 61.23 / 15.00 ≈ 4.08 inches
Case 2: w = 6.00
Similarly, we calculate the semiperimeter as:
s = (10.0 + 10.0 + 2 * 6.00) / 2 = 16.00 inches
The area of the triangle can be calculated as:
A = sqrt(s * (s - a) * (s - b) * (s - c))
= sqrt(16.00 * (16.00 - 10.0) * (16.00 - 10.00) * (16.00 - 10.00))
= sqrt(16.00 * 6.00 * 6.00 * 6.00)
= sqrt(3456.00)
≈ 58.82 square inches
The radius of the inscribed circle is:
r = A / s = 58.82 / 16.00 ≈ 3.68 inches
Case 3: w = 8.00
We calculate the semiperimeter as:
s = (10.0 + 10.0 + 2 * 8.00) / 2 = 18.00 inches
The area of the triangle can be calculated as:
A = sqrt(s * (s - a) * (s - b) * (s - c))
= sqrt(18.00 * (18.00 - 10.0) * (18.00 - 10.00) * (18.00 - 10.00))
= sqrt(18.00 * 8.00 * 8.00 * 8.00)
= sqrt(9216.00)
≈ 96.00 square inches
The radius of the inscribed circle is:
r = A / s = 96.00 / 18.00 ≈ 5.33 inches
To find the expression for the inscribed radius r in terms of the variable w, we can use the Pythagorean theorem in the triangle formed by the radius, the height of the triangle, and half of the base:
w^2 = r^2 + (10/2)^2
w^2 = r^2 + 25
r^2 = w^2 - 25
r = sqrt(w^2 - 25)
To find the value of w that gives the maximum value of r, we can take the derivative of r with respect to w, set it equal to zero, and solve for w:
dr/dw = 2w / (2 * sqrt(w^2 - 25)) = 0
w = sqrt(25)
w = 5.00 (rounded to the nearest hundredth)
Therefore, the value of w that gives the maximum value of r is approximately 5.00 inches.