Answered

Looking for answers? Westonci.ca is your go-to Q&A platform, offering quick, trustworthy responses from a community of experts. Ask your questions and receive accurate answers from professionals with extensive experience in various fields on our platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.

A restaurant's revenue history for the first 10 years in business is modeled by the function:

[tex]\[ R=0.0001\left(-t^4+12 t^3-77 t^2+600 t+13,050\right) \][/tex]

where [tex]\( t \)[/tex] is the number of years since the restaurant opened and [tex]\( R \)[/tex] is annual revenue in millions of dollars.

In which year(s) did the restaurant's revenue equal \[tex]$1.5 million?

The restaurant's revenue equals \$[/tex]1.5 million for the first time in year _______ and again in year _______.

Sagot :

GinnyAnswer
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.
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.