https://file.aiquickdraw.com/mathbot/file/1727211984090ar6xomvp.jpg

Let's analyze the graph and problem based on the image you've uploaded.

### a) Which of the lines A, B, or C shows the best estimate for the tangent to the curve at 10 years?
To estimate the tangent to a curve, we look for the line that most closely follows the direction of the curve at the specific point (in this case, 10 years). In the graph, we observe three lines—A, B, and C—passing close to the curve at this point.

Looking at 10 years:
- **Line B** is the line that touches the curve closest at 10 years, following the curve's direction most accurately at that point.
  
So, **Line B** provides the best estimate for the tangent.

### b) Use your answer to part a) to work out an estimate for the rate of change of money in the account at 10 years (to 2 decimal places).
The rate of change at a point on a curve is represented by the slope of the tangent line at that point. Since Line B is the tangent at 10 years, we calculate the slope of Line B as:

\[
\text{slope} = \frac{\text{change in money}}{\text{change in time}} = \frac{\Delta y}{\Delta x}
\]

From the graph:
- At 10 years (point where Line B intersects), the value on the y-axis is approximately 25 units of money.
- At around 20 years, Line B appears to intersect the y-axis at approximately 45 units of money.

So, the slope of Line B is:

\[
\text{slope} = \frac{45 - 25}{20 - 10} = \frac{20}{10} = 2
\]

Thus, the rate of change of money in the account at 10 years is approximately **2 units per year**.

If you'd like more details or have any questions, feel free to ask! Here are five related questions to further explore this topic:

1. How is the tangent line used to estimate the instantaneous rate of change in various scenarios?
2. How do we calculate the slope of a line between two points?
3. What does the slope of a tangent line represent in the context of real-life problems like interest or investments?
4. Can we estimate the rate of change at any other points along the curve using different tangents?
5. How does the slope change as the steepness of a line increases or decreases?

**Tip:** When calculating slopes, remember that a steeper line means a higher rate of change, while a flatter line means a lower rate of change.

The graph below models how the amount of money in a bank account increases over time.

 a) Which of the lines A, B, or C shows the best estimate for the tangent to the curve at 10 years?

 b) Use your answer to part a) to work out an estimate for the rate of change of money in the account at 10 years to 2 d.p.

Learn how to estimate the tangent to a curve at a specific point and calculate the rate of change of money in a bank account over time. This problem focuses on understanding the slope and tangent concepts, with a specific example using a bank account growth model. Suitable for students in Grades 9-11.

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