The problem involves identifying the initial amount $a$, the rate of growth $r$ as a percentage, and evaluating the exponential function $y = 350(1 + 0.75)^t$ for $t = 5$. Let’s solve this step by step:

Step 1: Initial amount $a$

The initial amount $a$ is the coefficient of the base in the exponential function. Here: $a = 350$

Step 2: Growth rate $r$

The growth rate $r$ is derived from the base of the exponential expression $(1 + 0.75)$. The $0.75$ represents the growth rate in decimal form. To convert it into a percentage: $r = 0.75 \times 100 = 75\%$

Step 3: Evaluate $y$ when $t = 5$

We substitute $t = 5$ into the equation: $y = 350(1 + 0.75)^5$ Simplify $(1 + 0.75)$ to $1.75$, and calculate: $y = 350(1.75)^5$

Using a calculator: $1.75^5 \approx 17.44$ $y = 350 \times 17.44 \approx 6104.0$

Final Answer:

1. $a = 350$
2. $r = 75\%$
3. When $t = 5$, $y \approx 6104.0$

Would you like further details on any step?

Related Questions:

1. What would happen to the growth rate if $r$ was $-0.75$?
2. How does the function change if $t$ is doubled?
3. Can this exponential function model population growth effectively? Why or why not?
4. What is the doubling time for this function?
5. How do you interpret the base $(1 + r)$ in real-world terms?

Tip:

Always double-check calculations when working with powers and large growth rates to ensure precision.