The problem involves finding the initial value $k$ and the growth factor $a$ for the exponential model $f(t) = k \cdot a^t$, given two points: $(4, 20)$ and $(6, 40)$.

Step 1: Set up the equations

From the exponential model $f(t) = k \cdot a^t$, we can substitute the given points into the equation:

For the point $(4, 20)$: $20 = k \cdot a^4$

For the point $(6, 40)$: $40 = k \cdot a^6$

Step 2: Solve for $a$

To eliminate $k$, divide the second equation by the first: $\frac{40}{20} = \frac{k \cdot a^6}{k \cdot a^4}$ $2 = a^2$ $a = \sqrt{2}$

Step 3: Solve for $k$

Now substitute $a = \sqrt{2}$ back into one of the original equations. Using the first equation: $20 = k \cdot (\sqrt{2})^4$ Since $(\sqrt{2})^4 = 4$: $20 = k \cdot 4$ $k = 5$

Final Answer:

• $k = 5$
• $a = \sqrt{2}$

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How do you solve exponential models if given three points?
2. What is the significance of the growth factor $a$ in real-world scenarios?
3. How does changing $k$ affect the graph of the exponential function?
4. Can exponential models handle decay situations, and how would $a$ change?
5. How do you verify solutions for exponential equations?

Tip: Always check your final values by substituting them back into the original equations to ensure accuracy.