Math Problem Statement
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. Question content area bottom Part 1 a) Find the value k of the exponential growth rate. Assume Upper V Subscript oequals211. kequals
enter your response here (Round to the nearest thousandth.)
Solution
To find the value of the exponential growth rate, we can use the formula for exponential growth:
Where:
- is the value of the item at time ,
- is the initial value of the item (in 1979, when ),
- is the exponential growth rate, and
- is the time that has passed since the initial value was recorded.
Given:
- The initial value in 1979,
- The value in 1985, and
- The time years.
Now, let's plug the values into the formula and solve for .
To solve for , we follow these steps:
- Divide both sides by 211:
- Take the natural logarithm (ln) of both sides:
- Solve for :
Let me calculate that for you.The value of the exponential growth rate is approximately , rounded to the nearest thousandth.
Would you like further details or have any questions?
Here are some related questions to explore:
- How can the exponential growth formula be used to project future values?
- What would the doll's value be in the year 1990 with the same growth rate?
- How does exponential growth differ from linear growth in terms of long-term value predictions?
- How do we interpret the growth rate in practical terms?
- What would the value of the doll have been in 1975 using reverse calculation?
Tip: Always check if the growth rate makes sense by comparing projected values with actual historical data to ensure accuracy in modeling.
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Math Problem Analysis
Mathematical Concepts
Exponential Growth
Logarithms
Formulas
V(t) = V_0 e^{kt}
k = \frac{1}{t} \ln \left( \frac{V(t)}{V_0} \right)
Theorems
Exponential Growth Formula
Suitable Grade Level
Grades 10-12
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