A toy tractor sold for ​$226 in 1980 and was sold again in 1988 for $ 492. Assume that the growth in the value V of the​ collector's item was exponential.
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b) Find the exponential growth function in terms of​ t, where t is the number of years since 1980.
​V(t)equals

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To find the exponential growth function, we start with the formula for exponential growth:

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

where:
- $ V(t) $ is the value of the toy tractor at time $ t $.
- $ V_0 $ is the initial value (at $ t = 0 $).
- $ e $ is the base of the natural logarithm.
- $ k $ is the growth constant.
- $ t $ is the time in years since 1980.

### Step 1: Identify the Initial Value
The initial value of the toy tractor in 1980 was $ V_0 = 226 $.

### Step 2: Set Up the Exponential Equation for 1988
In 1988, the toy tractor was worth $ 492 $. The number of years since 1980 is $ t = 8 $. Substituting these values into the formula:

\[
492 = 226 \cdot e^{8k}
\]

### Step 3: Solve for $ k $
1. Divide both sides by 226:

\[
\frac{492}{226} = e^{8k}
\]

2. Calculate the ratio:

\[
2.177 = e^{8k}
\]

3. Take the natural logarithm on both sides to solve for $ k $:

\[
\ln(2.177) = 8k
\]

4. Simplify:

\[
k = \frac{\ln(2.177)}{8}
\]

### Step 4: Calculate $ k $
Let’s compute the value of $ k $.

The growth constant $ k \approx 0.0972 $.

### Step 5: Exponential Growth Function
Now that we have $ k $, the exponential growth function $ V(t) $ can be written as:

\[
V(t) = 226 \cdot e^{0.0972t}
\]

### Final Answer
The exponential growth function is:

\[
V(t) = 226 \cdot e^{0.0972t}
\]

Would you like to see any additional details or have any questions? Here are five related questions for further exploration:

1. How would the function change if the initial value in 1980 were different?
2. What would the value of the tractor be in 1995 using this model?
3. How long would it take for the tractor's value to double?
4. How does changing the value of $ k $ affect the growth rate?
5. What is the value of $ k $ for other exponential models?

**Tip:** When dealing with exponential growth, remember that a small change in the growth constant $ k $ can lead to large changes in the final value over time.

A toy tractor sold for ​$226 in 1980 and was sold again in 1988 for $492. Assume that the growth in the value V of the​ collector's item was exponential. Find the exponential growth function in terms of​ t, where t is the number of years since 1980.

Explore how to find the exponential growth function for a toy tractor that increased in value from $226 in 1980 to $492 in 1988. This problem uses the exponential growth formula and logarithms to determine the growth rate, suitable for Grades 10-12.

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