A quadrilateral is called cyclic if its corners lie on the circumference of a circle. An example is shown
below left: the circle has radius 5
The area A of the quadrilateral for different x values is plotted on the right. It can be calculated that
this curve has derivative:
dA/dx=48x^2 + (x3 − 50x)*sqrt(100−x^2)−2400/very complicated equation
(a) Show that the quadrilateral has maximum area when x=sqrt(50)
(b) At which x values does the quadrilateral have minimum area?
(c) The curve has minima at x = −6 and x = −8. What do these minima mean, in terms of the area of the quadrilateral?
To answer these questions, we will first analyze the given derivative of the area, which is shown as:
dA/dx = 48x^2 + (x^3 - 50x)*sqrt(100 - x^2) - 2400
(a) To find the x value at which the quadrilateral has maximum area, we need to find the critical points of the area function. These occur when the derivative is equal to zero. Therefore, we set dA/dx = 0 and solve for x:
48x^2 + (x^3 - 50x)*sqrt(100 - x^2) - 2400 = 0
At this point, solving the equation becomes a bit complex and is not directly provided. However, you can use numerical methods or approximation techniques to find the root of this equation. Once you find the value of x that satisfies this equation, you will have the x-coordinate for the maximum area. In this case, when x = sqrt(50), the quadrilateral has maximum area.
(b) To determine the x values at which the quadrilateral has minimum area, we need to find the points where the derivative changes sign from positive to negative or vice versa. These points are called critical points. One way to find the critical points is to set the derivative equal to zero and solve for x:
48x^2 + (x^3 - 50x)*sqrt(100 - x^2) - 2400 = 0
However, since the equation for the derivative is very complicated, finding the exact values may not be straightforward. Again, you can utilize numerical methods or approximation techniques to estimate the x values that satisfy the equation. These x values will correspond to the x-coordinates of the minimum area for the quadrilateral.
(c) The given information states that the curve of the derivative has minima at x = -6 and x = -8. In terms of the area of the quadrilateral, these minima indicate that the rate of change of the area is the smallest at these x values. In other words, the area is changing at the slowest rate at x = -6 and x = -8. This does not necessarily mean that the area of the quadrilateral is at its minimum at these points, but rather, it indicates those specific values of x where the rate of change of the area is minimized.