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A = 2x*sqrt(36-x^2)
Domain would be -6<=x<=6
A rectangle is bounded by the x-axis and the semicircle y = ¡Ì36 ¨C x2, as shown in the figure below. Write the area A of the rectangle as a function of x, and determine the domain of the area function.
Steve
answered13 years ago
Explain Bot
answered11 months ago
To find the area of the rectangle, we need to determine the length of its sides.
First, let's look at the figure. We have a rectangle bounded by the x-axis and a semicircle with equation y = √(36 - x^2).
The rectangle's base is parallel to the x-axis and its height corresponds to the difference in y-values between the x-axis and the semicircle.
The base of the rectangle is the same as the x-axis, so its length is simply x.
To find the height, we need to compute the difference between the y-coordinate of the semicircle at x and the value of the x-axis at x.
The y-coordinate of the semicircle at x is given by y = √(36 - x^2).
The value of the x-axis at x is simply 0.
Therefore, the height of the rectangle is √(36 - x^2) - 0 = √(36 - x^2).
Now that we have the length and height of the rectangle, we can find its area by multiplying these two values together:
A(x) = length × height = x × √(36 - x^2).
The domain of the area function is determined by the restrictions on x imposed by the problem. In this case, x can vary from the leftmost point of the semicircle to the rightmost point of the semicircle.
The equation of the semicircle is y = √(36 - x^2). To find the leftmost and rightmost points, we set y = 0:
0 = √(36 - x^2)
0 = 36 - x^2
x^2 = 36
x = ±√36
x = ±6
Therefore, the domain of the area function is -6 ≤ x ≤ 6, or in interval notation, [-6, 6].