- Define the sales function \(s = 16\sqrt{3t} + 25\) with \(s\) in thousands.
- Create a table of values for weeks \(t = 0, 3, 7, 12\) to observe the growth trend.
- Substitute \(t = 7\) into the equation to get \(s \approx 98.3216\) (in thousands).
- Multiply by 1000 to find the total sales is about \(\boxed{\$98,000}\).
Explanation
- Problem Analysis
The sales of a product are modeled by the function \(s = 16\sqrt{3t} + 25\), where \(s\) is the total sales in thousands of dollars and \(t\) is the time in weeks after release. Our objective is to estimate the sales after 7 weeks by generating a table of values, considering the graph of this non-linear function, and finally calculating the value for \(t = 7\).
- Table of Values and Function Behavior
To understand the behavior of the function \(s(t) = 16\sqrt{3t} + 25\), we construct a table of values for several time points after the release:
| Week (\(t\)) | Sales (\(s\)) in thousands |
| :--- | :--- |
| 0 | \(16\sqrt{0} + 25 = 25\) |
| 3 | \(16\sqrt{9} + 25 = 16(3) + 25 = 73\) |
| 7 | \(16\sqrt{21} + 25 \approx 16(4.58) + 25 \approx 73.32 + 25 = 98.32\) |
| 12 | \(16\sqrt{36} + 25 = 16(6) + 25 = 121\) |
The function represents a square root curve that grows slowly over time. Plotting these points on a coordinate system shows a curve starting at \((0, 25)\) and increasing at a decreasing rate.
- Calculation and Conclusion
To find the sales after \(7\) weeks, we substitute \(t = 7\) into the given equation:
\[s = 16\sqrt{3(7)} + 25\]
\[s = 16\sqrt{21} + 25\]
Given that \(\sqrt{21} \approx 4.5826\), we perform the calculation:
\[s \approx 16(4.5826) + 25\]
\[s \approx 73.3216 + 25 = 98.3216\]
Since the value \(s\) is in thousands, the total sales in dollars is \(98.3216 \times 1000 \approx \$98,321.60\). Comparing this to the given options, we find that it is closest to \$98,000.
Examples
This mathematical model is essential for businesses to forecast growth and revenue. For example, if a retail store releases a new product, they can use functions like this to estimate inventory requirements or adjust marketing budgets based on anticipated sales trends over time, ensuring resources are allocated efficiently. Understanding how to interpret such growth models helps in making informed financial and operational decisions.