Question

A rectangle is inscribed under an arc of the graph of y = cos x.

Your goal is to find the largest possible area for this rectangle.
a. Give the coordinates (x, y) of the vertices of the rectangle
that touch the arc in terms of x.
b. In terms of x, what is the length of the rectangle?
C. In terms of x, what is the width of the rectangle?
d. Find an expression f(x) for the area of the rectangle in
terms of x.
e. What interval of x-values should be considered when
finding the area?
f. Find f(0.2), f(1), and xplain what each means in the
context of the problem.
g. Graph fover the domain you found in Part e, and find the
maximum value of the rectangle.

Answer 1

The largest area of a rectangle under y = cos x is found by using the vertices on the arc, determining length as 2x, width as cos x, and the area function as f(x) = 2x × cos x. The values of f(0.2) and f(1) represent the areas of the rectangles at x = 0.2 and x = 1. Graphing f(x) from 0 to π/2 reveals the maximum rectangle area.

Step-by-step explanation:

To determine the largest possible area of a rectangle inscribed under an arc on the graph of y = cos x, consider the following steps:

• The vertices of the rectangle that touch the arc will be at points (x, cos x) and (-x, cos x).
• The length of the rectangle is the distance along the x-axis between these points, which is 2x.
• The width of the rectangle is the y-value of the arc, which is cos x.
• An expression for the area of the rectangle in terms of x is f(x) = 2x × cos x.
• The interval of x-values should be considered from 0 to π/2, as cos x is positive and decreasing on this interval, which will maximize the area.
• Calculating f(0.2), f(1), and understanding these values mean finding the area of the inscribed rectangle at those specific points x on the function y = cos x.
• To find the maximum area, graph f(x) over the domain from 0 to π/2 and look for the highest value on the graph. This value corresponds to the maximum area of the inscribed rectangle.

Answer 2

Step-by-step explanation:

a. The coordinates of the vertices are (x, y) and (-x, y). Since y = cos(x), we can write the coordinates in terms of x as (x, cos(x)) and (-x, cos(x)).

b. The length of the rectangle is the horizontal distance between the vertices, or 2x.

c. The width of the rectangle is the vertical distance between the vertices and the x-axis, or cos(x).

d. The area of the rectangle is width times length:

f(x) = 2x cos(x)

e. The width of the rectangle must be greater than 0, so:

cos(x) > 0

-π/2 < x < π/2

f. f(0.2) = 2(0.2) cos(0.2) = 0.392

f(1) = 2(1) cos(1) = 1.081

f(π/2) = 2(π/2) cos(π/2) = 0

f(x) is the area for each rectangle where x is half the rectangle's length.

g. Graph: desmos.com/calculator/f7xhwfy2dj

f(x) reaches a maximum value of 1.122.