If an isosceles triangle has a fixed perimeter P, find the dimensions that maximize that area. Show how you know that it is the maximum.
p = 2x+y
a = 1/2 y √(x^2 - (y/2)^2)
= 1/2 (p-2x) √(x^2 - ((p-2x)/2)^2)
= 1/4 (p-2x)√(4px-p^2)
da/dx = p(p-3x)/√(4px-p^2)
da/dx=0 when p = 3x
That is, when the triangle is equilateral. I expect you saw that coming, eh?
To find the dimensions of the isosceles triangle that maximize its area given a fixed perimeter P, we need to relate the dimensions of the triangle to its area.
Let's assume that the base of the triangle is b, and the two equal sides are each denoted as s. Since it is an isosceles triangle, the perimeter P can be expressed as:
P = b + 2s
The area of a triangle can be calculated using the formula:
Area = (base * height) / 2
For an isosceles triangle, the height h can be found by applying the Pythagorean theorem:
h = √(s² - (b/2)²)
Now, let's express the area in terms of s and b using the above formulas:
Area = (b * √(s² - (b/2)²)) / 2
To find the maximum area, we need to differentiate the area equation with respect to b, set it equal to 0, and solve for b:
d(Area)/db = (1/2) * (√(s² - (b/2)²) - (b² / √(s² - (b/2)²))) = 0
Simplifying the above equation will give us:
√(s² - (b/2)²) - (b² / √(s² - (b/2)²)) = 0
Squaring both sides, we have:
s² - (b/2)² = b² / (s² - (b/2)²)
Cross-multiplying and simplifying further, we get:
s⁴ = b²(b² + 4s²)
Expanding and rearranging terms:
s⁴ - 4s²b² - b⁴ = 0
Now, we can solve this equation for values of s in terms of b. To do that, we can consider s² as a variable (let's call it x) and rewrite the equation:
x² - 4b²x - b⁴ = 0
This is a quadratic equation in terms of x. Solving this equation will give us the possible values of x, which correspond to the possible values of s². From these values, we can determine the possible values of s, and then find the corresponding values of b using the perimeter equation P = b + 2s.
To find the maximum area, we need to find the critical points by taking the derivative of the area equation with respect to s and setting it equal to 0. Then, we need to check whether the critical points correspond to a maximum or minimum area by evaluating the second derivative.
By solving the quadratic equation and evaluating the critical points, we can determine the dimensions of the isosceles triangle that maximize its area given a fixed perimeter P.