A rectangle is bounded by the x-and y- axes and the graph of y = -1/2x + 4.
a.) Find the area of the rectangle as a function of x.
b.) Complete the following table
x area
1
2
4
6
7
c.) What is the domain of this function?
a) x will be the width of the rectangle, and y will be the height.
A = x*y = x(-1/2x+4)
b) plug in x = 1, 2... 7
c) [0,8] because that's where the function crosses the x and y axis
To find the area of the rectangle as a function of x, we need to first determine the dimensions of the rectangle.
The rectangle is bounded by the x-axis, y-axis, and the graph of the equation y = -1/2x + 4.
a.) Find the area of the rectangle as a function of x:
Since the rectangle is bounded by the x-axis and y-axis, the base of the rectangle will be equal to the value of x, and the height will be equal to the value of y on the graph.
To find the height (y), substitute the value of x in the equation y = -1/2x + 4:
y = -1/2x + 4
Now, the area of the rectangle can be found by multiplying the base (x) by the height (y):
Area = x * y
Substitute the value of y from the equation:
Area = x * (-1/2x + 4)
Simplify further to get the area as a function of x:
Area = (-1/2)x^2 + 4x
b.) To complete the table, we need to substitute the given values of x and calculate the corresponding areas:
x | area
---|-----
1 | [substitute 1 in the equation]
2 | [substitute 2 in the equation]
4 | [substitute 4 in the equation]
6 | [substitute 6 in the equation]
7 | [substitute 7 in the equation]
To calculate the area for each value of x, substitute the given values in the area function:
x = 1: Area = (-1/2)(1)^2 + 4(1)
x = 2: Area = (-1/2)(2)^2 + 4(2)
x = 4: Area = (-1/2)(4)^2 + 4(4)
x = 6: Area = (-1/2)(6)^2 + 4(6)
x = 7: Area = (-1/2)(7)^2 + 4(7)
By substituting the values, you can calculate the corresponding areas.
c.) The domain of the function represents all the possible values of x for which we can calculate the area.
In this case, since the base of the rectangle is determined by the value of x, the domain of the function is all real numbers. Therefore, the domain of the function is (-∞, ∞).