Final answer:
The equation that models the situation is 40 = 2(length + width) and 35 = length * width.
By solving these equations simultaneously, we find that the width of Marie's rectangle is approximately 10 + √65 feet.
Step-by-step explanation:
To solve this problem, we can use the formula for the area of a rectangle, which is length times width. Let's call the width of Marie's rectangle 'w'.
The length of the rectangle is not given, but we know that the perimeter of the rectangle is 40 meters.
The perimeter of a rectangle is given by the formula P = 2(length + width). Since we know the perimeter is 40 meters and the width is 'w', we can write the equation as:
40 = 2(length + w)
Now we can substitute the given information about the area into the equation.
The area of a rectangle is given by the formula A = length * width, and we know that the area is 35 square meters.
So we can write the equation as:
35 = length * w
Now we have two equations:
40 = 2(length + w) and 35 = length * w
We can use these two equations to solve for 'w', the width of Marie's rectangle. Let's start by solving the first equation for length:
40 = 2(length + w)
Divide both sides by 2:
20 = length + w
Subtract 'w' from both sides:
20 - w = length
Now we can substitute this value of length into the second equation:
35 = (20 - w) * w
Expand the equation:
35 = 20w - w^2
Rearrange the equation:
w^2 - 20w + 35 = 0
This is a quadratic equation. We can use the quadratic formula to solve for 'w':
w = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -20, and c = 35. Substituting these values into the quadratic formula:
w = (-(-20) ± √((-20)^2 - 4(1)(35))) / (2(1))
Simplifying:
w = (20 ± √(400 - 140)) / 2
w = (20 ± √260) / 2
w = (20 ± √(4 * 65)) / 2
w = (20 ± 2√65) / 2
w = 10 ± √65
So the possible values for 'w' are 10 + √65 and 10 - √65.
However, since the width cannot be negative, the width of Marie's rectangle is approximately 10 + √65 feet.