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?
do you have a graphic calculator
yes
a.) To find the area of the rectangle, we need to find the length and width of the rectangle. The length of the rectangle is the x-coordinate of the point where the graph intersects the x-axis, and the width is the y-coordinate of the point where the graph intersects the y-axis.
Since the graph intersects the x-axis when y = 0, we can substitute y = 0 into the equation y = -1/2x + 4 and solve for x:
0 = -1/2x + 4
1/2x = 4
x = 8
Therefore, the length of the rectangle is 8.
The graph intersects the y-axis when x = 0. Substituting x = 0 into the equation gives us the y-coordinate:
y = -1/2(0) + 4
y = 4
Therefore, the width of the rectangle is 4.
The formula for the area of a rectangle is length multiplied by width. Plugging in the values we found, the area of the rectangle is:
Area = length * width
Area = 8 * 4
Area = 32
So, the area of the rectangle as a function of x is A(x) = 32.
b.) To complete the table, we will substitute the given values of x into A(x) = 32 to find the corresponding areas:
x | area
1 | A(1) = 32
2 | A(2) = 32
4 | A(4) = 32
6 | A(6) = 32
7 | A(7) = 32
Since the area of the rectangle is constant regardless of the value of x, all the areas in the table will be 32.
c.) The domain of this function is the range of x-values for which the function is defined. In this case, the equation y = -1/2x + 4 is defined for all real numbers, so the domain is (-∞, ∞), which means it includes all real numbers.