Let’s solve this step by step.

Part a: Making an Exponential Model

We need to find an exponential model of the form: $C(t) = C_0 \cdot e^{kt}$ where $C_0$ is the initial cost, $k$ is the growth rate, and $t$ is the time in years since 1980.

Given data points:

• $t = 0$, $C = 140$
• $t = 5$, $C = 250$
• $t = 10$, $C = 410$
• $t = 15$, $C = 780$
• $t = 20$, $C = 1300$
• $t = 22$, $C = 1520$

To find the parameters $C_0$ and $k$, we start with the initial value $C_0 = 140$, since this is the cost at $t = 0$.

Next, we use the data points to find $k$. We can select two points and solve for $k$. Let’s use $(t = 0, C = 140)$ and $(t = 5, C = 250)$.

The model equation becomes: $250 = 140 \cdot e^{5k}$

Divide both sides by 140: $\frac{250}{140} = e^{5k}$

Calculating the left side: $1.7857 \approx e^{5k}$

Taking the natural logarithm of both sides: $\ln(1.7857) = 5k$ $k = \frac{\ln(1.7857)}{5} \approx 0.115$

So, the exponential model is: $C(t) = 140 \cdot e^{0.115t}$

Part b: Plotting the Data and the Exponential Model

To plot the data, we will use the original data points and also plot the exponential model $C(t) = 140 \cdot e^{0.115t}$. This involves calculating the model values for various values of $t$ and comparing them with the given data points.

Part c: Yearly Percentage Growth Rate

The yearly percentage growth rate can be calculated using the formula: $\text{Growth Rate} = (e^k - 1) \times 100$ Substituting $k = 0.115$: $\text{Growth Rate} = (e^{0.115} - 1) \times 100 \approx 12.2\%$

Part d: Expected Average Cost in 2020

To find the expected cost in 2020, we use $t = 40$ (since 2020 is 40 years after 1980) in our exponential model: $C(40) = 140 \cdot e^{0.115 \times 40}$ Calculating the exponent: $0.115 \times 40 = 4.6$ Thus: $C(40) = 140 \cdot e^{4.6} \approx 140 \cdot 99.48 \approx 13927$

The expected average cost of physics and chemistry periodicals in 2020 is approximately $13,927.

---

Do you want further details on any part of the solution?

Related Questions:

1. How would the model change if we used different data points to calculate $k$?
2. How does the choice of initial data point affect the accuracy of an exponential model?
3. What would the model predict for a year like 2030?
4. How can we use logarithmic transformation to linearize an exponential model?
5. What are some limitations of using an exponential model for predicting long-term trends?

Tip: When calculating exponential growth, using the natural logarithm helps solve for growth rates easily due to the properties of $e$.