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.
And 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.
In this case, we know that the triangle has two 10.0-inch sides and a 2w-inch side. The semiperimeter, s, can be calculated as the sum of all three sides divided by 2:
s = (10.0 + 10.0 + 2w) / 2
For the area, we can use Heron's formula:
A = √(s * (s - 10.0) * (s - 10.0) * (s - 2w))
Now, let's plug in the given values of w and calculate the radius of the inscribed circle for each case:
1) For w = 5.00:
s = (10.0 + 10.0 + 2 * 5.00) / 2 = 15.00
A = √(15.00 * (15.00 - 10.0) * (15.00 - 10.0) * (15.00 - 2 * 5.00)) ≈ 50.00
r = 50.00 / 15.00 ≈ 3.33 inches
2) For w = 6.00:
s = (10.0 + 10.0 + 2 * 6.00) / 2 = 16.00
A = √(16.00 * (16.00 - 10.0) * (16.00 - 10.0) * (16.00 - 2 * 6.00)) ≈ 60.00
r = 60.00 / 16.00 ≈ 3.75 inches
3) For w = 8.00:
s = (10.0 + 10.0 + 2 * 8.00) / 2 = 18.00
A = √(18.00 * (18.00 - 10.0) * (18.00 - 10.0) * (18.00 - 2 * 8.00)) ≈ 72.00
r = 72.00 / 18.00 = 4.00 inches
To write an expression for the inscribed radius r in terms of the variable w, we can substitute the values of A and s derived from the formulas above into the initial formula:
r = √((s * (s - 10.0) * (s - 10.0) * (s - 2w)) / (s))
Simplifying the expression, we get:
r = √((s - 10.0) * (s - 10.0) * (s - 2w))
To find the value of w that gives the maximum value of r, we can take the derivative of the expression for r with respect to w, set it equal to zero, and solve for w. However, since the expression for r is nonlinear and involves a square root, it may not be straightforward to find the exact value of w that maximizes r analytically.
One approach to find an approximate value for w that maximizes r is to plot the graph of r as a function of w and identify the peak point on the graph. Using a graphing calculator or software, you can plot the function and find the value of w corresponding to the maximum point on the graph.