Question

TEST Unit 3B
Aaron Medina
#15.
The amount of fertilizer in a landscaping company's warehouse decreases at a rate of 2 %2% per week. The amount of fertilizer in the warehouse was originally, 64 , 00064,000 cubic yards. Which function models the amount of fertilizer in cubic yards left after "w"w weeks?


"f" left parenthesis, "w" , right parenthesis equals 0.9 8 left parenthesis, 64000 , right parenthesis Superscript, "w" , Baseline
"f" left parenthesis, "w" , right parenthesis equals 64 , 000 left parenthesis, 0.9 8 , right parenthesis Superscript, "w" , Baseline
"f" left parenthesis, "w" , right parenthesis equals 64 , 000 left parenthesis, 1.0 2 , right parenthesis Superscript, "w" , Baseline
"f" left parenthesis, "w" , right parenthesis equals 64 , 000 left parenthesis, 1.9 8 , right parenthesis Superscript, "w" , Baseline

Answer 1

To determine the correct function that models the amount of fertilizer left after "w" weeks, we begin with the initial amount of fertilizer and account for the rate of decrease.

1. **Initial Amount**: 64,000 cubic yards.
2. **Rate of Decrease**: The fertilizer decreases by 2% per week. Therefore, after each week, 98% (100% - 2%) of the fertilizer remains.
- This percentage in decimal form is \(0.98\).
3. **Exponential Decay Model**: The amount of fertilizer left after \(w\) weeks can be expressed as:
\[
f(w) = \text{initial amount} \times (\text{decay factor})^{w}
\]
Substituting in the given values:
\[
f(w) = 64000 \times (0.98)^{w}
\]

Now we can analyze the answer choices provided:

- "f(w) = 0.98(64000)^{w}" - Incorrect; it incorrectly applies the initial amount.
- "f(w) = 64000(0.98)^{w}" - Correct; this matches our derived formula.
- "f(w) = 64000(1.02)^{w}" - Incorrect; the rate is increasing, not decreasing.
- "f(w) = 64000(1.98)^{w}" - Incorrect; does not reflect the decay.

The correct function that models the amount of fertilizer left after "w" weeks is

**f(w) = 64000(0.98)^{w}.**