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I see that you have uploaded an image of a web assignment. Let's address the mathematical problems visible in the screenshot.

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### Problem 1:
"A certain piece of machinery was purchased 4 years ago by Garland Mills for $500,000. Its present resale value is $300,000. Assuming that the machine's resale value decreases exponentially, what will it be 5 years from now?"

This is an exponential decay problem, and we can use the following formula for exponential decay:

\[
V(t) = V_0 \cdot e^{-kt}
\]

Where:
- $V(t)$ is the value at time $t$,
- $V_0$ is the initial value,
- $k$ is the decay constant,
- $t$ is the time.

1. We know the initial value ($V_0 = 500,000$) and the value after 4 years ($V(4) = 300,000$).
   
   Substituting into the decay formula:
   \[
   300,000 = 500,000 \cdot e^{-4k}
   \]
   Simplifying for $k$:
   \[
   \frac{300,000}{500,000} = e^{-4k}
   \]
   \[
   0.6 = e^{-4k}
   \]
   Taking the natural logarithm of both sides:
   \[
   \ln(0.6) = -4k
   \]
   \[
   k = -\frac{\ln(0.6)}{4}
   \]
   Calculate $k$.

2. Once you have $k$, use it to find the value after 5 years. That is, substitute $t = 5 + 4 = 9$ (since we're calculating the value 5 years after the present time):
   \[
   V(9) = 500,000 \cdot e^{-9k}
   \]

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### Problem 2:
"Phosphorus-32 ($^{32}_{15}P$) has a half-life of 14.2 days. If 100 g of this substance are present initially, find the amount $Q$ present after 4 days. (Round your growth constant to four decimal places.)"

This is also an exponential decay problem, but with half-life, we use a different form of the equation:

\[
Q(t) = Q_0 \cdot e^{-kt}
\]

Where:
- $Q(t)$ is the amount after time $t$,
- $Q_0 = 100$ g (initial amount),
- $k$ is the decay constant (which we need to find using the half-life).

1. The formula for the decay constant in terms of half-life is:
   \[
   k = \frac{\ln(2)}{t_{\text{half}}}
   \]
   Substituting the half-life of 14.2 days:
   \[
   k = \frac{\ln(2)}{14.2}
   \]
   Calculate $k$.

2. Once you have $k$, substitute $t = 4$ days into the decay equation to find $Q(4)$:
   \[
   Q(4) = 100 \cdot e^{-4k}
   \]

Finally, there is a follow-up question: "What amount will be left after 13.5 days?" Use the same formula, with $t = 13.5$, to calculate this.

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Let me know if you'd like detailed calculations or have any specific questions! Here are 5 related questions for further exploration:

1. What does the decay constant $k$ represent in an exponential decay model?
2. How would you model the exponential decay if the machine's value decreased by a constant percentage each year?
3. What happens to the value of $V(t)$ as $t$ approaches infinity?
4. How is half-life related to the decay constant in exponential decay problems?
5. If the half-life of Phosphorus-32 were different, how would that affect the decay constant?

**Tip:** In exponential decay problems, understanding the relationship between the decay constant and the half-life can make it easier to solve other related problems efficiently.

A certain piece of machinery was purchased 4 years ago by Garland Mills for $500,000. Its present resale value is $300,000. Assuming that the machine's resale value decreases exponentially, what will it be 5 years from now? Phosphorus-32 (32/15 P) has a half-life of 14.2 days. If 100 g of this substance are present initially, find the amount Q present after 4 days. What amount will be left after 13.5 days?

Solve an exponential decay problem involving a machinery resale value decreasing over time, and a half-life problem with Phosphorus-32. Learn how to apply key formulas, including V(t) = V0 * e^(-kt) and k = ln(2) / t_half, with step-by-step guidance. Suitable for Grades 11-12.

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