A mechanical engineering student wishes to inscribe a rectangle in a quarter circle of radius 2.00 cm. Determine the dimensions of the rectangle that will give it the greatest area.
length in cm =
height in cm =
My attempt on find the relationship between the rectangle and the quarter circle:
Let the base of the rectangle = b
let the height of the rectangle = h
Area of a a rectangle = b*h
Area of a circle =pie*R^2
Quarter of a circle = 1/4 pie R^2
am I on the right track......
is this the correct thought process....
Would it be safe to say that the the diagonal line in the rectangle could equal r = 2.
so I could use
b^2+ h^2 =r^2
solve for h
h= Square root ( r^2 -b^2)
Area = b^2 * Square root (r^2-b^2)
area = b*(r^2-b^2)^1/2 ....... ????
ok....
so is next step take the first derv of the area
Area of rect '= -1/2(4-b^2) + (4-2^2)^1/2
Yes, your thought process is correct! To find the dimensions of the rectangle that will give it the greatest area inscribed in a quarter circle, we can maximize the area of the rectangle by finding the relationship between the rectangle and the quarter circle.
First, let's draw a diagram to visualize the problem. Label the radius of the quarter circle as 2.00 cm. Since the rectangle is inscribed in the quarter circle, its diagonal will be the diameter of the circle, which is the same as twice the radius of the quarter circle, equal to 4.00 cm.
We can divide the rectangle into two right triangles with sides equal to the radius of the quarter circle. By using Pythagoras' theorem, we can determine the relationship between the base (b) and height (h) of the rectangle.
Using the Pythagorean theorem:
b^2 + h^2 = (2r)^2
b^2 + h^2 = 4
To find the dimensions of the rectangle that will give it the greatest area, we need to express the area of the rectangle in terms of one variable. Since you defined the base of the rectangle as b and the height as h, the area of the rectangle is given by:
Area = Base * Height
A = b * h
Now, we can express one variable in terms of the other by solving the Pythagorean theorem equation for either b or h. Let's solve it for h:
h^2 = 4 - b^2
h = sqrt(4 - b^2)
Substituting this expression for h in the area equation, we get:
A = b * sqrt(4 - b^2)
Now, we have the area of the rectangle (A) expressed in terms of the base (b). To find the maximum value of A, we can take the derivative of A with respect to b, set it equal to zero, and solve for b:
dA/db = sqrt(4 - b^2) - b^2 / sqrt(4 - b^2) = 0
Solving this equation will give us the value of b that maximizes the area of the rectangle. Once we find the value of b, we can substitute it back into the h equation to find the corresponding value of h.
Using calculus or algebraic methods like factoring or completing the square, we can solve the equation for b and find the dimensions of the rectangle that will give it the greatest area.
I hope this explanation helps! Let me know if you have further questions.