Roy has a patio that covers 434 ft ^2. He wants to increase his patio 196 ft ^2 by adding 4 ft to both the length and width. The length of the original patio is 3 ft more than twice the width. What is the length of the new patio
First, Area=Length*Width
Original Length=3+2*Width
434=Original Length*Original Width
434=(3+2*Width)*Width
434=3*Width+2*Width^2
I solved by graphing, you can do another way if you want.
Width is 14 ft. Original Length is 31 ft.
Add 4 and you get the answer: 35 feet
Ty!
Algebraic way:
original width = x
original length = 2x+3
x(2x+3) = 434
2x^2 + 3x - 434 = 0
(x - 14)(2x + 31) = 0
x = 14 or x = -31/2 , which I will reject
original width is 14 ft
original length is 31 ft
new width = 18
new length = 35
To find the length of the new patio, we need to first determine the dimensions of the original patio, and then calculate the dimensions of the new patio.
Let's assign variables to the dimensions:
- Length of the original patio = L
- Width of the original patio = W
From the given information, we can form two equations.
Equation 1: The area of the original patio is 434 ft^2.
Area of the original patio = Length × Width
L × W = 434 (Equation 1)
Equation 2: The length of the original patio is 3 ft more than twice the width.
Length = 2 × Width + 3
To solve these equations, we can substitute the value of Length from Equation 2 into Equation 1.
Substituting the value of Length into Equation 1:
(2 × Width + 3) × Width = 434
Simplifying the equation:
2W^2 + 3W = 434
Rearranging the equation to standard quadratic form:
2W^2 + 3W - 434 = 0
Now, we can solve this quadratic equation to find the value of W. Once we have the value of W, we can substitute it back into Equation 2 to find the value of L. Finally, we can calculate the length of the new patio by adding 4 ft to both the length and width of the original patio.