A doll sold for ​$211 in 1979 and was sold again in 1985 for $ 439. Assume that the growth in the value V of the​ collector's item was exponential.
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Part 1
​a) Find the value k of the exponential growth rate. Assume Upper V Subscript oequals211.
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​(Round to the nearest​ thousandth.)

To find the value $ k $ of the exponential growth rate, we can use the formula for exponential growth:

\[
V(t) = V_0 e^{kt}
\]

Where:
- $ V(t) $ is the value of the item at time $ t $,
- $ V_0 $ is the initial value of the item (in 1979, when $ t = 0 $),
- $ k $ is the exponential growth rate, and
- $ t $ is the time that has passed since the initial value was recorded.

Given:
- The initial value $ V_0 = 211 $ in 1979,
- The value $ V(t) = 439 $ in 1985, and
- The time $ t = 1985 - 1979 = 6 $ years.

Now, let's plug the values into the formula and solve for $ k $.

\[
439 = 211 e^{6k}
\]

To solve for $ k $, we follow these steps:

1. Divide both sides by 211:

\[
\frac{439}{211} = e^{6k}
\]

2. Take the natural logarithm (ln) of both sides:

\[
\ln \left( \frac{439}{211} \right) = 6k
\]

3. Solve for $ k $:

\[
k = \frac{1}{6} \ln \left( \frac{439}{211} \right)
\]

Let me calculate that for you.The value of the exponential growth rate $ k $ is approximately $ 0.122 $, rounded to the nearest thousandth.

Would you like further details or have any questions?

Here are some related questions to explore:

1. How can the exponential growth formula be used to project future values?
2. What would the doll's value be in the year 1990 with the same growth rate?
3. How does exponential growth differ from linear growth in terms of long-term value predictions?
4. How do we interpret the growth rate $ k $ in practical terms?
5. What would the value of the doll have been in 1975 using reverse calculation?

**Tip:** Always check if the growth rate $ k $ makes sense by comparing projected values with actual historical data to ensure accuracy in modeling.

A doll sold for $211 in 1979 and was sold again in 1985 for $439. Assume that the growth in the value V of the collector's item was exponential. Find the value k of the exponential growth rate. Assume V0 = 211. k = ? (Round to the nearest thousandth.)

Solve an exponential growth problem where a doll's value grew from $211 in 1979 to $439 in 1985. Learn how to calculate the exponential growth rate (k) using logarithms and the exponential growth formula. Suitable for students in Grades 10-12.

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