find the perimeter of the rectangle with maximum area that can be inscribed in a semicircle of radius 2 ft. if two of its vertices are on the diameter of the semicircle and the other two are on the semicircle.
Did you make a sketch?
draw a line from the centre to a vertex of the rectangle lying on the semicircle.
Let the length of the rectangle be 2x and its height y
I see a right-angled triangle with sides x and y and hypotenuse 2
x^2 + y^2 = 4
let the area be A
A = 2xy
A^2 = 4x^2y^2 = 4x^2(4-x^2)
= 16x^2 - 4x^4
2A dA/dx = 32x - 16x^3 = 0 for max of A
16x(2 - x^2) = 0
x^2=2
x=√2
sub back into equation above
y = √2
so length = 2√2 and width = √2
Perimeter = 6√2
thank you so muchh :)
To find the perimeter of the rectangle with maximum area inscribed in a semicircle, we can follow these steps:
Step 1: Identify the dimensions of the rectangle.
- Let's assume the length of the rectangle is parallel to the diameter of the semicircle, and its width is perpendicular to the diameter.
- Let the length of the rectangle be 2x, and width be y.
Step 2: Express the area of the rectangle in terms of x.
- Since the area of a rectangle is given by length multiplied by width, the area A of the rectangle is given by A = 2x * y.
Step 3: Express the width y in terms of x using the Pythagorean theorem.
- The width y can be found by using the Pythagorean theorem, considering the radius of the semicircle as the hypotenuse of a right triangle.
- The radius (r) of the semicircle is given as 2 ft.
- The width y can be expressed as y = √(r^2 - x^2).
Step 4: Substitute the value of y in terms of x into the area equation.
- Substituting y = √(r^2 - x^2) into A = 2x * y gives A = 2x * √(r^2 - x^2).
Step 5: Maximize the area of the rectangle using differentiation.
- Differentiate A with respect to x and set the derivative equal to zero to find critical points.
- Find the second derivative to analyze if the critical points are maximum or minimum.
- Find the maximum value of A within the domain.
Step 6: Calculate the length of the rectangle (2x) and use it to find the perimeter of the rectangle.
- Substitute the value of x that maximizes the area back into 2x to find the length of the rectangle.
- The perimeter of the rectangle is given by P = 2(2x + y).
By following these steps, you can find the dimensions and perimeter of the rectangle with maximum area that can be inscribed in the semicircle.