Question

The area of a rectangular garden is given by the quadratic function:A(x)=-6x^2+105x-294A . Knowing that the area, length, and width all must be a positive value puts restrictions on the value of x. What is the domain for the function? Explain how you determined the domain. For what value of x, produces the maximum area? What is the maximum area of the garden? What is the Range of the function? Explain how you determined the range? What value(s) of x produces an area of 100 square units?

Answer 1

Answer and Step-by-step explanation:

The domain of a function is the values the invariable can assume to result in a real value for the variable. In other words, it is all the values x can be.

Since it's related to area, the values of x has to be positive. The domain must be, then:

`-6x^(2) + 105x - 294 = 0`

Solving the second degree equation:

`\frac{-105+\sqrt{105^(2) - 4(-2)(-294)} }{2(-6)}`

x = 3.5 or x = 14

The domain of this function is 3.5 ≤ x ≤ 14

The maximum area is calculated by taking the first derivative of the function:

`(dA)/(dx) = -6x^(2) + 105x - 294`

A'(x) = -12x + 105

-12x + 105 = 0

-12x = -105

x = 8.75

A(8.75) =
`-6.8.75^(2) + 105.8.75 - 294`

A(8.75) = 165.375

The maximum area of the garden is 165.375 square units.

The Range of a function is all the value the dependent variable can assume. So, the range of this function is: 0 ≤ y ≤ 165.375, since this value is the maximum it will reach.

A(x) = 100

`100 = -6x^(2) + 105x-294`

`-6x^(2) + 105x - 394 = 0`

Solving:

`\frac{-105+\sqrt{105^(2)-4(-6)()-394} }{2(-6)}`

x = 5.45 or x = 12.05

The values of x that produces an area of 100 square units are 5.45 and 12.05