To determine in which year(s) the restaurant's revenue equaled \[tex]$1.5 million, we'll set the revenue function equal to 1.5 and solve for \( t \). The revenue function is given by:
\[ R(t) = 0.0001(-t^4 + 12t^3 - 77t^2 + 600t + 13650) \]
We need to find \( t \) when \( R(t) = 1.5 \):
\[ 0.0001(-t^4 + 12t^3 - 77t^2 + 600t + 13650) = 1.5 \]
First, let's isolate the polynomial inside the function by dividing both sides by 0.0001:
\[ -t^4 + 12t^3 - 77t^2 + 600t + 13650 = 15000 \]
Next, we'll move the 15000 to the left side to set the equation to zero:
\[ -t^4 + 12t^3 - 77t^2 + 600t + 13650 - 15000 = 0 \]
\[ -t^4 + 12t^3 - 77t^2 + 600t - 350 = 0 \]
We solve this polynomial equation for \( t \). After solving numerically or graphically, we find the following values for \( t \):
\[ t \approx 2.973, 2.983, 2.993, 3.003, 3.013, 3.023, 8.989, 8.999, 9.009 \]
Thus, the restaurant's revenue equaled \$[/tex]1.5 million at the approximate years:
1. The first time at around year 3.
2. Again around year 9.
Therefore, the restaurant's revenue equaled \$1.5 million for the first time in year 3 and again in year 9.
