Respuesta :
From the question;
we are given the function
[tex]p(x)=-45x^{3^{}^{}}+2500x^2\text{ - 275,000}[/tex]where
p(x) = profit
x = cost of advertising
we are to find the smallest of the two advertising amounts that produce a profit of $800,000.
this implies
p(x) = $800, 000
Therefore we have
[tex]800,000=-45^3+2500x^2\text{ - 275,000}[/tex]by simplifying the equation we get
[tex]\begin{gathered} 45x^3-2500x^2\text{ + 800,000 + 275,000 = 0} \\ 45x^3-2500x^2\text{ + 1,075,000 = 0} \end{gathered}[/tex]solving the equation using a calculator
we get the values of x to be
[tex]\begin{gathered} x_{1_{}}=-18.0,x_2=42.0_{} \\ \text{and } \\ x_3=\text{ 31.5} \end{gathered}[/tex]Since cost cannot be negative,
then the real solutions are
[tex]\begin{gathered} x_2\text{ = 42.0 } \\ \text{and } \\ x_3=32.0\text{ ( to the nearest whole number)} \end{gathered}[/tex]Therefore,
The smaller of the two advertising amounts that produce a profit of $800,000 is
x = 32
