Respuesta :
Answer:
The values of x for which the model is 0 ≤ x ≤ 3
Step-by-step explanation:
The given function for the volume of the shipping box is given as follows;
V = 2·x³ - 19·x² + 39·x
The function will make sense when V ≥ 0, which is given as follows
When V = 0, x = 0
Which gives;
0 = 2·x³ - 19·x² + 39·x
0 = 2·x² - 19·x + 39
0 = x² - 9.5·x + 19.5
From an hint obtained by plotting the function, we have;
0 = (x - 3)·(x - 6.5)
We check for the local maximum as follows;
dV/dx = d(2·x³ - 19·x² + 39·x)/dx = 0
6·x² - 38·x + 39 = 0
x² - 19/3·x + 6.5 = 0
x = (19/3 ±√((19/3)² - 4 × 1 × 6.5))/2
∴ x = 1.288, or 5.045
At x = 1.288, we have;
V = 2·1.288³ - 19·1.288² + 39·1.288 ≈ 22.99
V ≈ 22.99 in.³
When x = 5.045, we have;
V = 2·5.045³ - 19·5.045² + 39·5.045≈ -30.023
Therefore;
V > 0 for 0 < x < 3 and V < 0 for 3 < x < 6.5
The values of x for which the model makes sense and V ≥ 0 is 0 ≤ x ≤ 3.
The values of x for which the model makes sense are 13/2 and 3
The equation that represents the volume of the box is given as:
[tex]V =2x\³ -19x\² +39x[/tex]
Factor out x from the above equation
[tex]V =x(2x^2 -19x +39)[/tex]
Expand the above equation, as follows
[tex]V =x(2x^2 -6x - 13x +39)[/tex]
Factorize the above equation
[tex]V =x(2x(x -3) - 13(x -3))[/tex]
Factor out x - 3
[tex]V =x(2x -13)(x -3)[/tex]
Equate to 0
[tex]x(2x -13)(x -3) = 0[/tex]
Solve for x
[tex]x = 0[/tex] or [tex]x = 13/2[/tex] or [tex]x = 3[/tex]
The value of x cannot be 0.
Hence, the values of x for which the model makes sense are 13/2 and 3
Read more about volumes at:
https://brainly.com/question/1972490
