The given exponential function is:

$y = 575 \left( 1 - 0.6 \right)^t$

To identify the initial amount $a$ and the rate of decay $r$, we first recognize the structure of an exponential function:

$y = a \left( 1 - r \right)^t$

Here, $a$ is the initial amount, and $1 - r$ represents the decay factor.

• The initial amount $a$ is the coefficient in front of the exponential term, which is 575.
• The rate of decay $r$ is the quantity that is subtracted from 1. Since we have $1 - 0.6$, the rate of decay $r$ is 0.6, or 60% (as a percent).

Now, evaluating the function at $t = 3$:

$y = 575 \left( 1 - 0.6 \right)^3 = 575 \left( 0.4 \right)^3$

Calculating $0.4^3$:

$0.4^3 = 0.064$

Now multiply by 575:

$y = 575 \times 0.064 = 36.8$

Thus, when $t = 3$, the value of $y$ is 36.8 (rounded to the nearest tenth).

Summary:

• Initial amount $a = 575$
• Rate of decay $r = 60\%$
• Value at $t = 3$ is $y = 36.8$

Would you like further clarification on any of these steps? Here are some related questions to explore:

1. How do you interpret the rate of decay in real-life scenarios (e.g., population decline, radioactive decay)?
2. How would the result change if the decay rate were higher (e.g., 80%)?
3. What would the value of $y$ be if $t = 0$?
4. How would you solve for $t$ if you are given a specific value for $y$?
5. How does the decay factor affect the long-term behavior of the function?

Tip: When working with exponential decay, note that the decay factor is always less than 1, which causes the value of the function to decrease as $t$ increases.