Question
How do you make snow withdrawals after three months do you have $150 after six months he has $300 after nine months he has $150 what is the rate of change
Answers
GPT-4o mini
To find the rate of change from the information you provided about snow withdrawals, we can look at the changes in the amounts over time.
Let's summarize the amounts after each specified time period:
- After 3 months: $150
- After 6 months: $300
- After 9 months: $150
Now, we'll calculate the rate of change between each time period.
1. **From 3 months to 6 months**:
\[
\text{Change} = 300 - 150 = 150
\]
\[
\text{Time Change} = 6 - 3 = 3 \text{ months}
\]
\[
\text{Rate of Change} = \frac{150}{3} = 50 \, \text{dollars per month}
\]
2. **From 6 months to 9 months**:
\[
\text{Change} = 150 - 300 = -150
\]
\[
\text{Time Change} = 9 - 6 = 3 \text{ months}
\]
\[
\text{Rate of Change} = \frac{-150}{3} = -50 \, \text{dollars per month}
\]
### Summary of Rates of Change:
- From 3 months to 6 months: +$50 per month
- From 6 months to 9 months: -$50 per month
These results indicate that there was a positive rate of change (increase) from 3 to 6 months, and a negative rate of change (decrease) from 6 to 9 months.
Let's summarize the amounts after each specified time period:
- After 3 months: $150
- After 6 months: $300
- After 9 months: $150
Now, we'll calculate the rate of change between each time period.
1. **From 3 months to 6 months**:
\[
\text{Change} = 300 - 150 = 150
\]
\[
\text{Time Change} = 6 - 3 = 3 \text{ months}
\]
\[
\text{Rate of Change} = \frac{150}{3} = 50 \, \text{dollars per month}
\]
2. **From 6 months to 9 months**:
\[
\text{Change} = 150 - 300 = -150
\]
\[
\text{Time Change} = 9 - 6 = 3 \text{ months}
\]
\[
\text{Rate of Change} = \frac{-150}{3} = -50 \, \text{dollars per month}
\]
### Summary of Rates of Change:
- From 3 months to 6 months: +$50 per month
- From 6 months to 9 months: -$50 per month
These results indicate that there was a positive rate of change (increase) from 3 to 6 months, and a negative rate of change (decrease) from 6 to 9 months.
So is the answer -50
GPT-4o mini
The answer of -50 dollars per month represents the rate of change between 6 months and 9 months, where the amount decreased from $300 to $150.
However, if you're looking for a single overall rate of change for the entire time period (from 3 months to 9 months), you'd calculate the change from the initial amount at 3 months to the amount at 9 months.
### Overall Rate of Change from 3 months to 9 months:
1. **Initial amount at 3 months**: $150
2. **Final amount at 9 months**: $150
\[
\text{Overall Change} = 150 - 150 = 0
\]
\[
\text{Time Change} = 9 - 3 = 6 \text{ months}
\]
\[
\text{Overall Rate of Change} = \frac{0}{6} = 0 \, \text{dollars per month}
\]
### Conclusion:
For the overall rate of change from 3 months to 9 months, the answer is **0 dollars per month**.
But specifically, between 6 months and 9 months, the rate of change is **-50 dollars per month**. So it depends on which specific time period you are referring to for the rate of change.
However, if you're looking for a single overall rate of change for the entire time period (from 3 months to 9 months), you'd calculate the change from the initial amount at 3 months to the amount at 9 months.
### Overall Rate of Change from 3 months to 9 months:
1. **Initial amount at 3 months**: $150
2. **Final amount at 9 months**: $150
\[
\text{Overall Change} = 150 - 150 = 0
\]
\[
\text{Time Change} = 9 - 3 = 6 \text{ months}
\]
\[
\text{Overall Rate of Change} = \frac{0}{6} = 0 \, \text{dollars per month}
\]
### Conclusion:
For the overall rate of change from 3 months to 9 months, the answer is **0 dollars per month**.
But specifically, between 6 months and 9 months, the rate of change is **-50 dollars per month**. So it depends on which specific time period you are referring to for the rate of change.
Find the rate of change of Pete’s height from 3 to 5 years
Time (years) 1,2,3,4,5, hight(in.) 27, 35, 37, 42, 45, 49
Time (years) 1,2,3,4,5, hight(in.) 27, 35, 37, 42, 45, 49
GPT-4o mini
To find the rate of change of Pete's height from 3 to 5 years, we need the height values for those years. Here are the height values given:
- At 3 years: 37 inches
- At 5 years: 45 inches
### Calculate the Change in Height
1. **Change in height**:
\[
\text{Height at 5 years} - \text{Height at 3 years} = 45 \, \text{inches} - 37 \, \text{inches} = 8 \, \text{inches}
\]
### Calculate the Change in Time
2. **Change in time**:
\[
\text{Time at 5 years} - \text{Time at 3 years} = 5 \, \text{years} - 3 \, \text{years} = 2 \, \text{years}
\]
### Calculate the Rate of Change
3. **Rate of change**:
\[
\text{Rate of Change} = \frac{\text{Change in Height}}{\text{Change in Time}} = \frac{8 \, \text{inches}}{2 \, \text{years}} = 4 \, \text{inches per year}
\]
### Conclusion
The rate of change of Pete's height from 3 to 5 years is **4 inches per year**.
- At 3 years: 37 inches
- At 5 years: 45 inches
### Calculate the Change in Height
1. **Change in height**:
\[
\text{Height at 5 years} - \text{Height at 3 years} = 45 \, \text{inches} - 37 \, \text{inches} = 8 \, \text{inches}
\]
### Calculate the Change in Time
2. **Change in time**:
\[
\text{Time at 5 years} - \text{Time at 3 years} = 5 \, \text{years} - 3 \, \text{years} = 2 \, \text{years}
\]
### Calculate the Rate of Change
3. **Rate of change**:
\[
\text{Rate of Change} = \frac{\text{Change in Height}}{\text{Change in Time}} = \frac{8 \, \text{inches}}{2 \, \text{years}} = 4 \, \text{inches per year}
\]
### Conclusion
The rate of change of Pete's height from 3 to 5 years is **4 inches per year**.