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Sagot :
Let's analyze the problem step by step:
### Objective:
We need to determine the years [tex]\( t \)[/tex] (since the restaurant opened) when the restaurant's revenue [tex]\( R(t) \)[/tex] is equal to [tex]$1.5$[/tex] million dollars.
### Given Revenue Function:
[tex]\[ R(t) = 0.0001(-t^4 + 12t^3 - 77t^2 + 600t + 13,050) \][/tex]
### Revenue Target:
The target revenue is [tex]$1.5$[/tex] million, which we can write as:
[tex]\[ R(t) = 1,500,000 \][/tex]
### Setting Up the Equation:
We equate the revenue function to [tex]$1.5$[/tex] million.
[tex]\[ 0.0001(-t^4 + 12t^3 - 77t^2 + 600t + 13,050) = 1,500,000 \][/tex]
To simplify calculations, we can multiply both sides of the equation by [tex]$10,000$[/tex] (this cancels out the [tex]$0.0001$[/tex] factor):
[tex]\[ -t^4 + 12t^3 - 77t^2 + 600t + 13,050 = 15,000,000 \][/tex]
### Rearrange the Equation:
We rearrange the equation to set it to zero, as this forms a polynomial that we can solve for roots:
[tex]\[ -t^4 + 12t^3 - 77t^2 + 600t + 13,050 - 15,000,000 = 0 \][/tex]
This further simplifies to:
[tex]\[ -t^4 + 12t^3 - 77t^2 + 600t - 14,986,950 = 0 \][/tex]
### Solving the Polynomial:
To find the values of [tex]\( t \)[/tex], we solve the polynomial equation:
[tex]\[ -t^4 + 12t^3 - 77t^2 + 600t - 14,986,950 = 0 \][/tex]
### Results:
Upon solving this polynomial equation, we find that there are no real and positive solutions for [tex]\( t \)[/tex].
### Conclusion:
Since there are no positive, real solutions for [tex]\( t \)[/tex], the restaurant's revenue never reaches exactly [tex]$1.5$[/tex] million at any point during the first 10 years.
So, the answer is:
The restaurant's revenue does not equal [tex]$1.5$[/tex] million in any year within the timeframe given.
### Objective:
We need to determine the years [tex]\( t \)[/tex] (since the restaurant opened) when the restaurant's revenue [tex]\( R(t) \)[/tex] is equal to [tex]$1.5$[/tex] million dollars.
### Given Revenue Function:
[tex]\[ R(t) = 0.0001(-t^4 + 12t^3 - 77t^2 + 600t + 13,050) \][/tex]
### Revenue Target:
The target revenue is [tex]$1.5$[/tex] million, which we can write as:
[tex]\[ R(t) = 1,500,000 \][/tex]
### Setting Up the Equation:
We equate the revenue function to [tex]$1.5$[/tex] million.
[tex]\[ 0.0001(-t^4 + 12t^3 - 77t^2 + 600t + 13,050) = 1,500,000 \][/tex]
To simplify calculations, we can multiply both sides of the equation by [tex]$10,000$[/tex] (this cancels out the [tex]$0.0001$[/tex] factor):
[tex]\[ -t^4 + 12t^3 - 77t^2 + 600t + 13,050 = 15,000,000 \][/tex]
### Rearrange the Equation:
We rearrange the equation to set it to zero, as this forms a polynomial that we can solve for roots:
[tex]\[ -t^4 + 12t^3 - 77t^2 + 600t + 13,050 - 15,000,000 = 0 \][/tex]
This further simplifies to:
[tex]\[ -t^4 + 12t^3 - 77t^2 + 600t - 14,986,950 = 0 \][/tex]
### Solving the Polynomial:
To find the values of [tex]\( t \)[/tex], we solve the polynomial equation:
[tex]\[ -t^4 + 12t^3 - 77t^2 + 600t - 14,986,950 = 0 \][/tex]
### Results:
Upon solving this polynomial equation, we find that there are no real and positive solutions for [tex]\( t \)[/tex].
### Conclusion:
Since there are no positive, real solutions for [tex]\( t \)[/tex], the restaurant's revenue never reaches exactly [tex]$1.5$[/tex] million at any point during the first 10 years.
So, the answer is:
The restaurant's revenue does not equal [tex]$1.5$[/tex] million in any year within the timeframe given.
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