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 kk of the exponential growth rate, we can use the formula for exponential growth:

V(t)=V0ektV(t) = V_0 e^{kt}

Where:

  • V(t)V(t) is the value of the item at time tt,
  • V0V_0 is the initial value of the item (in 1979, when t=0t = 0),
  • kk is the exponential growth rate, and
  • tt is the time that has passed since the initial value was recorded.

Given:

  • The initial value V0=211V_0 = 211 in 1979,
  • The value V(t)=439V(t) = 439 in 1985, and
  • The time t=19851979=6t = 1985 - 1979 = 6 years.

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

439=211e6k439 = 211 e^{6k}

To solve for kk, we follow these steps:

  1. Divide both sides by 211:

439211=e6k\frac{439}{211} = e^{6k}

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

ln(439211)=6k\ln \left( \frac{439}{211} \right) = 6k

  1. Solve for kk:

k=16ln(439211)k = \frac{1}{6} \ln \left( \frac{439}{211} \right)

Let me calculate that for you.The value of the exponential growth rate kk is approximately 0.1220.122, 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 kk 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 kk 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