- Define the function \(s = 10\sqrt{5t} + 15\) for sales in thousands.
- Calculate values at key intervals, specifically finding \(s \approx 78.25\) at \(t = 8\).
- Recognize that \(78.25\) in thousands represents approximately \$78,250.
- Conclude that the correct option is \(\boxed{\text{about \$78,000}}\).
Explanation
- Problem Analysis
The sales function is given by \(s = 10\sqrt{5t} + 15\), where \(s\) is the total sales in thousands and \(t\) is the time in weeks after the product release. Our objective is to determine the total sales 8 weeks after the release by creating a table of values, graphing the function, and evaluating the expression for \(t = 8\).
- Table of Values
To understand the behavior of the sales over time, we calculate the sales at various time points using the function \(s = 10\sqrt{5t} + 15\).
| Time \(t\) (weeks) | Sales \(s\) (thousands) |
| :--- | :--- |
| 0 | \(10\sqrt{0} + 15 = 15\) |
| 2 | \(10\sqrt{10} + 15 \approx 46.62\) |
| 4 | \(10\sqrt{20} + 15 \approx 59.72\) |
| 6 | \(10\sqrt{30} + 15 \approx 69.77\) |
| 8 | \(10\sqrt{40} + 15 \approx 78.25\) |
| 10 | \(10\sqrt{50} + 15 \approx 85.71\) |
These values help us visualize the growth of the product's sales, which increases rapidly at first and then more gradually over time.
- Graphing and Estimation
Using the values calculated above, we can plot the points on a graph where the x-axis represents time \(t\) and the y-axis represents sales \(s\). The graph follows the square root function pattern, starting at \((0, 15)\) and passing through \((8, 78.25)\). When we look at the graph at \(t = 8\), the corresponding y-value is approximately 78.25. Since \(s\) is in thousands, the total sales are approximately \(78.25 \times 1,000 = 78,250\), which is approximately \$78,000.
- Final Calculation and Conclusion
The calculation \(s = 10\sqrt{40} + 15\) gives approximately 78.25. Given that \(s\) is measured in thousands, the total sales at 8 weeks is approximately 78.25 thousand dollars, which is approximately \$78,000. Comparing this with the provided choices, the correct option is about \$78,000. \(\boxed{\text{about \$78,000}}\)
Examples
This type of function is useful in business to model product life cycles. By understanding how a product's sales grow over time—often following patterns like square root or logarithmic curves—managers can predict future revenue, plan inventory levels, and evaluate the success of marketing campaigns. For example, if a clothing retailer releases a new seasonal collection, they might use similar models to estimate how much stock they need for the following months based on initial sales performance.