I just need to know how to set this problem up.
You are given 28 feet of fencing and need to enclose a patio on all four sides. The following variables are defined:
Let X represent the width (in feet) and Y represent the length (in feet)of the enclosed patio. Let P represent the perimeter (in feet) and A represent the area(in feet)of the enclosed patio.
A. If the enclosed patio is 48 square feet, use the defined variables above to determine the two equations you would need to solve for x and y.
Eq1: A = x * y = 48ft^2.
Eq2: P = 2x + 2y = 28 Ft.
To set up the problem, we need to use the given variables and equations to find the dimensions of the enclosed patio.
The perimeter (P) of the patio can be calculated by summing up all four sides of the fence. Since we are given 28 feet of fencing, the equation for the perimeter is:
P = 2X + 2Y
The area (A) of the patio is given as 48 square feet. The equation for the area is:
A = XY
Now, let's solve for X and Y using these two equations.
From the perimeter equation, we can rearrange it to solve for X:
2X + 2Y = P
Divide both sides by 2:
X + Y = P/2
Subtract Y from both sides:
X = (P/2) - Y
Now, substitute this value for X in the area equation:
A = XY
48 = [(P/2) - Y] * Y
Expand the equation:
48 = (P/2)Y - Y^2
Subtract 48 from both sides:
0 = -Y^2 + (P/2)Y - 48
This quadratic equation can now be solved using methods such as factoring, completing the square, or the quadratic formula. The solutions obtained will provide the possible values for Y (length) and subsequently allow us to find X (width) by using the equation X = (P/2) - Y.
By solving the quadratic equation above, you will obtain the two equations needed to determine the values of X and Y, given the perimeter (28 feet) and area (48 square feet) of the enclosed patio.