Respuesta :
Answer:
The lengths of each side = 1 mile
[tex]A=\frac{x}{4}\sqrt{9-6x}[/tex]
The largest area = 1/4(√3) miles²
Step-by-step explanation:
∵ Th length of the base = x
∵ Perimeter of Δ = 3 miles
∵ The triangle is isosceles
∴ The length of the two equal sides = (3 - x)/2
∵ The height of the Δ is ⊥ to the base and bisect it
∴ h² = [(3 - x)/2]² - (x/2)² = (9 - 6x + x²)/4 - x²/4
∴ h² = 9/4 - 3/2 x + x²/4 - x²/4 = 9/4 - 3/2 x
∴ h = √(9/4 - 3/2 x) = √[(9 - 6x)/4] = 1/2[√(9 - 6x)]
∴ Area of the Δ = (1/2) (x) [1/2√(9 - 6x)]
∴ A = x/4 [√(9 - 6x) ]
∴ [tex]A=\frac{x}{4}\sqrt{9-6x}[/tex]
∵ The largest area of the given perimeter is the area
of the equilateral triangle
∴ The length of each side = 3/3 = 1 mile
∴ The largest area = 1/4(√3) miles²
The area is given by half the length of the base times the height, which
gives a length of 1 yard for the base and 1 yard for each equal side.
Response:
- [tex]Area \ as \ a \ function \ of \ x \ is; = \dfrac{x}{4} \times \sqrt{ {9-6\cdot x}[/tex]
- The lengths of the sides of the triangle that gives the largest area is 1 yard each
How can the length of the sides that gives the maximum area be found?
The given parameter are;
The shape of the region = Isosceles
The length of the fencing = 3 miles
The length of the base of the triangle = x
Required:
The lengths of the sides of the triangle that gives the maximum area.
Solution:
3 = x + 2·l
Which gives;
[tex]Length \ of \ the \ equal \ sides, \, l = \mathbf{\dfrac{3 - x}{2}}[/tex]
The height of the triangle according to Pythagorean theorem is therefore;
[tex]h^2 = \left( \dfrac{3 - x}{2} \right)^2 - \left(\dfrac{x}{2}\right)^2 = \mathbf{\dfrac{9-6\cdot x}{4}}[/tex]
Which gives;
[tex]Area \ of \ the \ region, \, A = \dfrac{x}{2} \times \sqrt{ \dfrac{9-6\cdot x}{4}} = \dfrac{x}{4} \times \sqrt{ {9-6\cdot x}[/tex][tex]Area \ of \ the \ region, \, A = \dfrac{x}{4} \times \sqrt{ {9-6\cdot x}[/tex]
The area of the region as a function of x is therefore;
[tex]Area \ of \ the \ region, \, A = \dfrac{x}{4} \times \sqrt{ {9-6\cdot x}[/tex]
Therefore;
Maximum or minimum (extremum) area are given as follows;
At the extremum point, we have;
[tex]\dfrac{dA}{dx} = \dfrac{d}{dx} \left(\dfrac{x}{4} \times \sqrt{ {9-6\cdot x}\right) = \mathbf{\dfrac{(3\cdot \sqrt{3} \cdot x - 3\cdot \sqrt{3} )\cdot \sqrt{3-2\cdot x }}{2\cdot x - 12}} = 0[/tex]
Which gives;
3·√3·x = 3·√3
At the maximum point, x = 1
Therefore;
[tex]Length \ of \ the \ equal \ sides, \, l = \dfrac{3 - 1}{2} = 1[/tex]
- The lengths of the three sides that give maximum area is 1 yard each
Learn more about the maximum value of a function here:
https://brainly.com/question/494897
