An isosceles trapezoid is inscribed under the graph oof y =cos x. what are the dimensions which will give the greatest area?
I will assume you want the base to be on the x-axis.
Let's make the base as long as possible, that is, from (-pi/2,0) to (pi/2,0)
Let the other point of contact be (x,y)
Area of trap = (1/2)(pi + 2x)(y)
= (1/2)(pi+2x)cosx = (1/2)picosx + xcosx
d(Area)/dx = (-1/2)pi(sinx) + x(-sinx) + cosx
= 0 for a max/min of Area
-pi(sinx) - 2xsinx + 2cosx = 0
2cosx = sinx(pi + 2x)
2/(pi + 2x) = tanx
I ran this through a primitive Newton's Method program and got
x = .45797
impossible
To find the dimensions that will give the greatest area for an isosceles trapezoid inscribed under the graph of y = cos x, we can break down the problem into a few steps:
Step 1: Understand the problem
An isosceles trapezoid has two parallel sides of equal length. In this case, since it is inscribed under the graph of y = cos x, the two parallel sides will be parallel to the x-axis.
Step 2: Find the base length
Since the two parallel sides are equal, let's call their length 2a. To find the value of a, we need to find the x-values where the graph of y = cos x intersects the x-axis.
Since cos x = 0 when x = (2n + 1)π/2, where n is an integer, we can see that the graph intersects the x-axis at these values. For simplicity, let's consider the interval from 0 to 2π. In this range, cos x = 0 at x = π/2 or x = 3π/2.
Since the trapezoid is isosceles, we can take the distance between these two values as the base length. So, the base length is (3π/2 - π/2) = π.
Step 3: Find the height
The height of the trapezoid is the distance between the x-axis and the highest point on the graph within the interval we are considering (0 to 2π).
Since cos x is periodic with a period of 2π, we can observe that the maximum value of cos x within this interval is 1. Thus, the height of the trapezoid is 1.
Step 4: Calculate the area
The formula for the area of a trapezoid is given by: Area = (base1 + base2) * height / 2.
In this case, since the trapezoid is isosceles, both bases will have a length of π.
Plugging in the values, the area is: Area = (π + π) * 1 / 2 = π.
Therefore, the dimensions that give the greatest area for the isosceles trapezoid are a base length of π and a height of 1, resulting in an area of π.