A rectangular plot of land that has an area of 1200 sq. ft, will hold a rectangular garden whose length is 8' more than twice the width, and will be bordered by a 3' wide sidewalk for pedestrians. Write a function that describes the area of the garden with walkway. What will the dimensions of the garden area be?
w = width of garden
a = (w + 3 + 3) (2 w + 8 + 3 + 3) = (w + 6) (2 w + 14)
... = 2 (w^2 + 13 w + 42) = 1200
solve for w ... w^2 + 13 w - 558 = 0
Thank you, how'd you get -558 though?
Oh nevermind, I figured it out!
To solve this problem, we need to find the dimensions of the garden area, including the sidewalk.
Let's start by defining some variables:
Let the width of the garden be represented by 'x'.
The length of the garden is given as 8 feet more than twice the width, which can be expressed as (2x + 8).
The width of the sidewalk is 3 feet.
The area of the garden can be calculated by multiplying the length and width of the garden:
Area of the garden = Length * Width
However, since we need to include the sidewalk, we need to add the width of the sidewalk on each side (remember, the sidewalk goes all the way around the garden). So the total width of the garden, including the sidewalk, is:
Total width = Width of garden + Width of sidewalk + Width of sidewalk
Since the sidewalk is on both sides of the garden, we add the width of the sidewalk twice.
Total width = x + 3 + 3 = (x + 6)
Similarly, the total length of the garden, including the sidewalk, is:
Total length = Length of garden + Width of sidewalk + Width of sidewalk
Total length = (2x + 8) + 3 + 3 = (2x + 14)
Now, we can calculate the area of the garden, including the sidewalk, by multiplying the total length and the total width:
Area of the garden with walkway = Total length * Total width
Area of the garden with walkway = (2x + 14) * (x + 6)
Therefore, the function that describes the area of the garden with the walkway is:
f(x) = (2x + 14) * (x + 6)
To find the dimensions of the garden area, we need to find the value of 'x' that maximizes the area. We can do this by taking the derivative of the function f(x) and setting it equal to zero. This will give us the value of 'x' at which the function reaches its maximum.
After solving for 'x', we can substitute this value back into the equation (2x + 8) for the length and x for the width to find the dimensions of the garden area.