a rectangle is inscribed in the upper half of the circle x^2+y^2=a^2 calculate the area of the largest such rectangle.
So A=2xy
y=sqrt(a^2-x^2)A
what do I do from there?
A = 2 x (a^2+x^2)^.5
dA/dx = 2 x (2x)(.5)(a^2+x^2)^-.5) + 2 (a^2+x^2)^.5
= 0 for max or min
the answer is a^2
To find the area of the largest rectangle inscribed in the upper half of the circle, you need to maximize the area function A = 2xy, where y = sqrt(a^2 - x^2) is the height of the rectangle.
To determine the maximum area, you can take the derivative of the area function A with respect to x and set it equal to zero to find the critical points. Let's compute the derivative:
dA/dx = 2y + 2x(dy/dx)
Now substitute the value of y from the equation y = sqrt(a^2 - x^2):
dA/dx = 2(sqrt(a^2 - x^2)) + 2x(dy/dx)
To find dy/dx, differentiate y = sqrt(a^2 - x^2) with respect to x:
dy/dx = -x / sqrt(a^2 - x^2)
Substitute dy/dx back into the expression for dA/dx:
dA/dx = 2(sqrt(a^2 - x^2)) + 2x(-x / sqrt(a^2 - x^2))
Now simplify:
dA/dx = 2sqrt(a^2 - x^2) - 2x^2 / sqrt(a^2 - x^2)
To find the critical points, set dA/dx equal to zero and solve for x:
2(sqrt(a^2 - x^2)) - 2x^2 / sqrt(a^2 - x^2) = 0
Multiplying through by sqrt(a^2 - x^2):
2(a^2 - x^2) - 2x^2 = 0
Simplifying:
2a^2 - 4x^2 = 0
Dividing by 2:
a^2 - 2x^2 = 0
Rearranging:
2x^2 = a^2
x^2 = a^2 / 2
Taking the square root:
x = +/- sqrt(a^2 / 2)
Since we are looking for a rectangle in the upper half of the circle, we can take the positive square root:
x = sqrt(a^2 / 2)
Now that we have the x-coordinate of the critical point, substitute it into the equation y = sqrt(a^2 - x^2) to find the y-coordinate:
y = sqrt(a^2 - (sqrt(a^2 / 2))^2)
Simplifying:
y = sqrt(a^2 - (a^2 / 2))
y = sqrt(a^2 / 2)
Now we have the values of x and y at the critical point. To find the area of the largest rectangle, substitute these values into the area formula A = 2xy:
A = 2(sqrt(a^2 / 2)) * sqrt(a^2 / 2)
Simplifying:
A = 2(a^2 / 2)
A = a^2
Therefore, the area of the largest rectangle inscribed in the upper half of the circle x^2 + y^2 = a^2 is a^2.