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 in the isosceles triangle, we can use the formula:
r = (a + b - c) / 2
where r is the radius of the inscribed circle, a and b are the lengths of the two equal sides of the triangle, and c is the length of the base side.
Given that the two equal sides of the triangle are 10.0 inches each and the length of the base side is 2w inches, we can substitute these values into the formula.
For w = 5.00:
a = b = 10.0 inches
c = 2w = 2 * 5.00 = 10.00 inches
Substituting these values into the formula:
r = (10.0 + 10.0 - 10.00) / 2
r = 10.0 / 2
r = 5.00 inches
For w = 6.00:
a = b = 10.0 inches
c = 2w = 2 * 6.00 = 12.00 inches
Substituting these values into the formula:
r = (10.0 + 10.0 - 12.00) / 2
r = 8.0 / 2
r = 4.00 inches
For w = 8.00:
a = b = 10.0 inches
c = 2w = 2 * 8.00 = 16.00 inches
Substituting these values into the formula:
r = (10.0 + 10.0 - 16.00) / 2
r = 4.0 / 2
r = 2.00 inches
Now, let's write an expression for the inscribed radius r in terms of w:
r = (10.0 + 10.0 - 2w) / 2
r = (20.0 - 2w) / 2
r = 10.0 - w
To find the value of w that gives the maximum value of r, we can take the derivative of r with respect to w and set it equal to 0.
dr/dw = -1
Setting dr/dw equal to 0, we have:
-1 = 0
Since this is not possible, it means that there is no maximum value of r.
Therefore, the maximum value of r would occur at the largest value of w, which is w = 8.00.
To find the radius of the inscribed circle of an isosceles triangle, we can use the following formula:
r = (s - w) / 2
Where:
- r is the radius of the inscribed circle
- s is the perimeter of the triangle, which can be calculated as s = 2s1 + s2, where s1 and s2 are the lengths of the two equal sides of the triangle
- w is the length of the side of the triangle that is not equal to the other two sides.
Let's calculate the radius of the inscribed circle for the given values of w:
Case 1: w = 5.00
In this case, the two equal sides are 10.0 inches, and the remaining side is 5.00 inches.
First, calculate the perimeter:
s1 = s2 = 10.0 inches
s = 2s1 + s2 = 2(10.0) + 5.0 = 25.0 inches
Now, calculate the radius of the inscribed circle:
r = (s - w) / 2 = (25.0 - 5.00) / 2 = 10.0 inches
Case 2: w = 6.00
In this case, the two equal sides are again 10.0 inches, but the remaining side is 6.00 inches.
Again, calculate the perimeter:
s1 = s2 = 10.0 inches
s = 2s1 + s2 = 2(10.0) + 6.0 = 26.0 inches
Now, calculate the radius of the inscribed circle:
r = (s - w) / 2 = (26.0 - 6.00) / 2 = 10.0 inches
Case 3: w = 8.00
In this case, the two equal sides are still 10.0 inches, and the remaining side is 8.00 inches.
Calculate the perimeter:
s1 = s2 = 10.0 inches
s = 2s1 + s2 = 2(10.0) + 8.0 = 28.0 inches
Now, calculate the radius of the inscribed circle:
r = (s - w) / 2 = (28.0 - 8.00) / 2 = 10.0 inches
As you can see, regardless of the value of w, the radius of the inscribed circle remains constant at 10.0 inches.
Now, let's write an expression for the inscribed radius r in terms of the variable w:
r = (s - w) / 2
And to find the value of w that gives the maximum value of r, we can take the derivative of r with respect to w and set it equal to zero. This will give us the critical point where the maximum occurs:
dr/dw = -1/2
Setting -1/2 equal to zero:
-1/2 = 0
Since this equation has no solution, it means that the radius of the inscribed circle is constant, and there is no maximum or minimum value for r with respect to w.