Respuesta :
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:
[tex]-6x^{2} + 105x - 294 = 0[/tex]
Solving the second degree equation:
[tex]\frac{-105+\sqrt{105^{2} - 4(-2)(-294)} }{2(-6)}[/tex]
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:
[tex]\frac{dA}{dx} = -6x^{2} + 105x - 294[/tex]
A'(x) = -12x + 105
-12x + 105 = 0
-12x = -105
x = 8.75
A(8.75) = [tex]-6.8.75^{2} + 105.8.75 - 294[/tex]
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
[tex]100 = -6x^{2} + 105x-294[/tex]
[tex]-6x^{2} + 105x - 394 = 0[/tex]
Solving:
[tex]\frac{-105+\sqrt{105^{2}-4(-6)()-394} }{2(-6)}[/tex]
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
