Franco is building a rectangular skating rink at the community park. the materials available limit the perimeter of the skating rink at most 180 feet. Let X represent the width of the skating rink. let a represent the area of the skating rink. the function a(×)=-x squared+90x area of the skating rink and function of the width. determine the domain and range of the function.

To determine the domain and range of the function a(x) = -x^2 + 90x, we need to consider the restrictions and possibilities.

Domain:
The domain of a function represents all the possible input values, in this case, the width of the skating rink (x). However, there is a known restriction on the perimeter of the skating rink, which is at most 180 feet. The perimeter of a rectangle is calculated as P = 2(length + width), so we have the equation:

2(length + width) ≤ 180

To find the length in terms of the width, we can rearrange the equation:

length ≤ (180 - 2width) / 2
length ≤ 90 - x

In this scenario, we assume that the length should be positive. Therefore, the domain of the function is all possible width values (x) that satisfy:

x ≥ 0 (width should be non-negative)
length ≤ 90 - x

Range:
The range of a function represents all the possible output values (a), in this case, the area of the skating rink. As the equation for the area is a(x) = -x^2 + 90x, we need to consider the limitations of the function.

The coefficient of x^2 is negative, indicating that the graph of the function is a downward-opening parabola. This means that the function will have a maximum value at some point.

Since we are looking for the maximum area, we need to find the vertex of the parabola. The vertex can be found using the formula x = -b / (2a), where a and b are the coefficients of the quadratic equation.

For the equation a(x) = -x^2 + 90x, the coefficient of x^2 is -1, and the coefficient of x is 90. So, we have:

x = -90 / (2 * (-1))
x = -90 / -2
x = 45

So, the width (x) at which the maximum area occurs is 45 feet. Now, we can substitute this value into the equation to find the maximum area:

a(x) = -45^2 + 90(45)
a(x) = -2025 + 4050
a(x) = 2025

Therefore, the maximum area is 2025 square feet.

Now, considering the range, since the coefficient of the x^2 term is negative, the graph opens downward, and the maximum area occurs at x = 45, the range of the function is:

a ≤ 2025

Therefore, the range of the function is all possible area values (a) that satisfy:

a ≤ 2025

To determine the domain and range of the function, let's consider the given equation:

a(x) = -x^2 + 90x

Domain:
The domain of a function refers to the set of possible input values. In this case, the width of the skating rink, represented by "x," cannot be negative. Additionally, the perimeter of a rectangular skating rink is determined by adding the lengths of all sides, which means that the sum of the width and length should not exceed 180 feet.

To find the domain, we need to find the values of x that satisfy both conditions. Since the width cannot be negative, x must be greater than or equal to zero. Additionally, the sum of the width and length should not exceed 180 feet, which can be written as:

2(width + length) ≤ 180

Since we are only considering the width, the equation becomes:

2(x + length) ≤ 180
x + length ≤ 90

To ensure that the width and length satisfy this inequality, we can say that the width must be less than or equal to 90. Combining both conditions, we get the domain of the function:

Domain = {0 ≤ x ≤ 90}

Range:
The range of a function refers to the set of possible output values. In this case, the area of the skating rink is given by the equation:

a(x) = -x^2 + 90x

Since the coefficient of the quadratic term (-x^2) is negative, the function represents a downward opening parabola. This means that the range will be the set of all real numbers less than or equal to the maximum value of the function.

The maximum value of the quadratic function can be achieved at the vertex. The x-coordinate of the vertex can be calculated using the formula:

x = -b / (2a)

In this case, a = -1 and b = 90. Substituting these values into the formula, we get:

x = -90 / (2 * -1) = 45

To find the maximum value of the function, we substitute this x-coordinate into the equation:

a(45) = -(45)^2 + 90(45) = -2025 + 4050 = 2025

Therefore, the maximum value of the function is 2025.

Consequently, the range of the function will be:

Range = {-∞ ≤ a ≤ 2025}

To summarize:
Domain of the function: {0 ≤ x ≤ 90}
Range of the function: {-∞ ≤ a ≤ 2025}