a rectangle with a perimeter 70 feet can have an area of 150 ft^2. use A(x) to demonstrate this fact algebraically and graphically.
To demonstrate algebraically that a rectangle with a perimeter of 70 feet can have an area of 150 square feet, we can use the information given and form equations.
Let's denote the length of the rectangle as "x" and the width as "y". The perimeter of a rectangle is given by the formula: P = 2(x + y). We know the perimeter is 70 feet, so we can write the equation as:
70 = 2(x + y)
Now, let's look at the area of the rectangle. The area A of a rectangle is given by the formula: A = xy. We know the area is 150 square feet, so we can write the equation as:
150 = xy
Now we have a system of two equations:
Equation 1: 70 = 2(x + y)
Equation 2: 150 = xy
To graphically demonstrate this fact, we can plot the points that satisfy these equations on a coordinate plane. Since the equations are nonlinear, we won't get a straight line graph. Instead, we will plot points that satisfy both equations.
First, we can rearrange the first equation to solve for y:
70 = 2(x + y)
35 = x + y
y = 35 - x
Now we substitute this value of y into the second equation:
150 = xy
150 = x(35 - x)
150 = 35x - x^2
Rearranging this equation gives us a quadratic equation:
x^2 - 35x + 150 = 0
We can use this equation to find the x-values that satisfy both equations. By solving this quadratic equation, we get two possible values for x.
Once we have the values of x, we can substitute them back into Equation 1 to find the corresponding y-values. These points (x, y) will represent the dimensions of a rectangle with a perimeter of 70 feet and an area of 150 square feet.
You can use a graphing calculator or software to plot the points and see the graphical representation of these solutions.