The plan for a rectangular patio has a square garden centered in the patio with sides parallel to the sides of the patio. The patio is 20m long and 12m wide. If each side of the garden is "x" meters, the function, y= 240-x^2 gives the area of the patio in m^2. What values makes sense for the domain? Explain why
total area of patio = 240
area of garden = x^2
useable patio area = 12 - x^2
if x is more than 12, you can not walk around the garden
The width of 12 m governs the value of x
We must have
12 - 2x > 0
x < 6
the square garden must have sides less than 6 m each.
In order to determine the domain of a function, we need to consider the possible values that the independent variable, in this case, "x," can take.
In this situation, we are given that the rectangular patio is 20m long and 12m wide. The square garden is centered in the patio, with sides parallel to the sides of the patio. Let's analyze the dimensions of the garden.
Since the garden is square and centered in the patio, we can conclude that both the length and width of the garden should be equal. Let's call this side length "x."
From this information, we can deduce that the maximum value for "x" will be half of the length and half of the width of the patio. Therefore, the maximum value for "x" would be half of the smaller dimension, which is 12m/2 = 6m.
The minimum value for "x" would be zero since it wouldn't make any sense to have a negative or non-existent garden.
In summary, to find the domain, we need to consider the range of values that "x" can take. In this case, the values that make sense for the domain are 0 ≤ x ≤ 6.