Rectangle ABCD has vertices C and d on the x-axis and vertices A and B on part of the parabola y = 9 - x^2 that is above the x-axis.
a.) Find the coordinates of A, B, C, and D in terms of x.
b.) Determine the Area of the rectangle as a function of x.
c.) What is the domain for the area function? Hint: Area cannot be negative
well I'm completley lost here on how to do this
also for c. technically speaking can't area be negative area is a tensor a magnitude with more then one direciton no? So an area can be negative it would just be in the set of complex numbers no?
a.) To find the coordinates of the vertices A, B, C, and D, we need to find the values of x at which the parabola intersects the x-axis. This occurs when y = 0.
For point C, y = 0, so we have 0 = 9 - x^2. Solving for x, we get x^2 = 9, which gives us x = ±3. Therefore, point C has coordinates (3, 0).
For point D, we have the same equation as point C since it lies on the x-axis. So point D also has coordinates (-3, 0).
To find points A and B, we need to find the y-values corresponding to x = ±3 on the parabola. Substituting these values into the equation y = 9 - x^2, we get:
A: A = 9 - (3)^2 = 9 - 9 = 0
B: B = 9 - (-3)^2 = 9 - 9 = 0
Therefore, points A and B both have coordinates (±3, 0).
b.) To determine the area of the rectangle as a function of x, we need to find the length and width of the rectangle. The length is the distance between points A and B, which is 2x units, and the width is the distance between points C and D, which is 6 units.
Area = length × width = 2x × 6 = 12x.
So, the Area of the rectangle as a function of x is A(x) = 12x.
c.) The domain for the area function, A(x), represents all possible values of x for which the area is defined and non-negative. Given that the width of the rectangle is always positive (6 units), the area can never be negative.
Hence, the domain for the area function is all real numbers, or (-∞, +∞).
a.) To find the coordinates of the vertices A, B, C, and D, we need to determine the points where the parabola intersects with the x-axis and then use those x-values to find the corresponding y-values.
Since the equation of the parabola is y = 9 - x^2, we set y = 0 to find the x-intercepts:
0 = 9 - x^2
x^2 = 9
x = ±√9
x = ±3
So, the x-coordinates of C and D are -3 and 3, respectively.
To find the y-values for A and B, we substitute these x-coordinates back into the equation of the parabola:
For point A with x = -3:
y = 9 - (-3)^2
y = 9 - 9
y = 0
Thus, the coordinates of A are (-3, 0).
For point B with x = 3:
y = 9 - 3^2
y = 9 - 9
y = 0
Thus, the coordinates of B are (3, 0).
Therefore, the coordinates of the rectangle are:
A: (-3, 0)
B: (3, 0)
C: (-3, 0)
D: (3, 0)
b.) The area of a rectangle is given by the product of its length and width. In this case, the length of the rectangle is the difference in x-coordinates between points D and C, which is 3 - (-3) = 6 units. The width of the rectangle is the distance between the x-axis and the parabola, which is the y-value of the parabola at either point A or B, i.e., y = 0.
Therefore, the area of the rectangle, as a function of x, is:
Area(x) = Length(x) * Width(x)
Area(x) = 6 * 0
Area(x) = 0
c.) In this case, the domain for the area function is determined by the x-values at which the rectangle exists. From part a., we found that the x-coordinates of points C and D are -3 and 3, respectively. Thus, the domain for the area function is -3 ≤ x ≤ 3, as these are the values of x that define the rectangle.
No, in the context of this problem, we are considering the area of the rectangle in a two-dimensional Cartesian coordinate system. In this context, area cannot be negative.
Let's break down the problem step by step:
a.) Find the coordinates of A, B, C, and D in terms of x:
We know that vertices C and D lie on the x-axis, so their y-coordinates are 0. Therefore, C has coordinates (x, 0) and D has coordinates (x+w, 0), where w is the width of the rectangle.
Vertices A and B lie on the part of the parabola y = 9 - x^2 above the x-axis. To find their coordinates, we substitute y = 9 - x^2 into the equation and solve for x:
9 - x^2 = 0
x^2 = 9
x = ±√9
x = ±3
Therefore, A has coordinates (-3, 9 - (-3)^2) = (-3, 0) and B has coordinates (3, 9 - 3^2) = (3, 0).
b.) Determine the Area of the rectangle as a function of x:
The length of the rectangle is the difference between the x-coordinates of A and B: L = (3 - (-3)) = 6.
The width of the rectangle is the difference between the x-coordinates of C and D: w = x+w - x = w.
Therefore, the area of the rectangle is given by A = L * w = 6 * w.
c.) What is the domain for the area function?
The domain represents all possible values that x can take. In this case, x represents the x-coordinate of vertex C and can be any real number.
However, we need to consider that the parabola y = 9 - x^2 is only above the x-axis. Therefore, x cannot be greater than the values that make y=0. From the equation y = 9 - x^2, we find that when y = 0, x = ±√9 = ±3.
So the domain for the area function is all real numbers between -∞ and +∞, excluding the interval (-3, 3).
To summarize:
a.) The coordinates of A, B, C, and D are:
A: (-3, 0)
B: (3, 0)
C: (x, 0)
D: (x+w, 0)
b.) The area of the rectangle is A = 6 * w.
c.) The domain for the area function is all real numbers except for the interval (-3, 3).