Use the following formula. nt (¹)-1] + A= P () Suppose that you drive 50,000 miles per year and gas averages $4 per gallon. Complete parts a. and b. below. a. What will you save in annual fuel expenses by owning a hybrid car averaging 50 miles per gallon rather than an SUV averaging 25 miles per gallon? $ (Round to the nearest dollar as needed.) b. If you deposit your monthly fuel savings at the end of each month into an annuity that pays 4.8% compounded monthly, how much will have saved at the end of five years? $ (Round to the nearest dollar as needed.)
Use the following formula. nt (¹)-1] + A= P () Suppose that you drive 50,000 miles per year and gas averages $4 per gallon. Complete parts a. and b. below. a. What will you save in annual fuel expenses by owning a hybrid car averaging 50 miles per gallon rather than an SUV averaging 25 miles per gallon? $ (Round to the nearest dollar as needed.) b. If you deposit your monthly fuel savings at the end of each month into an annuity that pays 4.8% compounded monthly, how much will have saved at the end of five years? $ (Round to the nearest dollar as needed.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
4.1 Answer both questions
![### Educational Website Content
**Fuel Savings Calculation Using Financial Formula**
#### Formula:
\[
A = \frac{P \left( \left(1 + \frac{r}{n}\right)^{nt} - 1 \right)}{\frac{r}{n}}
\]
#### Scenario:
Suppose you drive 50,000 miles per year and gas averages $4 per gallon. Complete parts a. and b. below.
---
**a. Annual Fuel Savings Calculation**
- Calculate the annual savings by driving a hybrid car averaging 50 miles per gallon versus an SUV averaging 25 miles per gallon.
- **Calculation Steps**:
1. **Hybrid Car**:
- Miles per gallon = 50
- Annual gallons used = 50,000 miles / 50 mpg = 1,000 gallons
- Expense = 1,000 gallons * $4/gallon = $4,000
2. **SUV**:
- Miles per gallon = 25
- Annual gallons used = 50,000 miles / 25 mpg = 2,000 gallons
- Expense = 2,000 gallons * $4/gallon = $8,000
3. **Savings**:
- $8,000 (SUV) - $4,000 (Hybrid) = $4,000
- **Result**: $4,000 (round to the nearest dollar as needed)
**b. Annuity Future Value Calculation**
- If you deposit your monthly fuel savings at the end of each month into an annuity that pays 4.8% interest compounded monthly, calculate the amount saved at the end of five years.
- **Formula Application**:
- P = Monthly savings ($4,000 annual savings / 12 months)
- r = Annual interest rate (4.8%)
- n = Number of compounding periods per year (12)
- t = Total time in years (5)
- **Result**: Calculate using the provided formula and round to the nearest dollar as needed.
---
These calculations demonstrate how switching to a more fuel-efficient hybrid vehicle can result in significant annual fuel savings and further financial growth when invested wisely.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6030cbbc-dba6-4cc4-b2c7-5d66af95ec0b%2F647e4223-295a-4583-b8ed-886776448e87%2F6vas5qu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Website Content
**Fuel Savings Calculation Using Financial Formula**
#### Formula:
\[
A = \frac{P \left( \left(1 + \frac{r}{n}\right)^{nt} - 1 \right)}{\frac{r}{n}}
\]
#### Scenario:
Suppose you drive 50,000 miles per year and gas averages $4 per gallon. Complete parts a. and b. below.
---
**a. Annual Fuel Savings Calculation**
- Calculate the annual savings by driving a hybrid car averaging 50 miles per gallon versus an SUV averaging 25 miles per gallon.
- **Calculation Steps**:
1. **Hybrid Car**:
- Miles per gallon = 50
- Annual gallons used = 50,000 miles / 50 mpg = 1,000 gallons
- Expense = 1,000 gallons * $4/gallon = $4,000
2. **SUV**:
- Miles per gallon = 25
- Annual gallons used = 50,000 miles / 25 mpg = 2,000 gallons
- Expense = 2,000 gallons * $4/gallon = $8,000
3. **Savings**:
- $8,000 (SUV) - $4,000 (Hybrid) = $4,000
- **Result**: $4,000 (round to the nearest dollar as needed)
**b. Annuity Future Value Calculation**
- If you deposit your monthly fuel savings at the end of each month into an annuity that pays 4.8% interest compounded monthly, calculate the amount saved at the end of five years.
- **Formula Application**:
- P = Monthly savings ($4,000 annual savings / 12 months)
- r = Annual interest rate (4.8%)
- n = Number of compounding periods per year (12)
- t = Total time in years (5)
- **Result**: Calculate using the provided formula and round to the nearest dollar as needed.
---
These calculations demonstrate how switching to a more fuel-efficient hybrid vehicle can result in significant annual fuel savings and further financial growth when invested wisely.
![**Title: Understanding Savings and Annuity Calculations**
**Formula for Annuity:**
\[ A = P \left[ \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}} \right] \]
Where:
- \( A \) = the future value of the annuity
- \( P \) = the payment amount per period
- \( r \) = annual interest rate
- \( n \) = number of times interest is compounded per year
- \( t \) = number of years
**Scenario:**
Suppose you drive 50,000 miles per year and gas averages $4 per gallon. Calculate the following:
**a. Annual Fuel Savings:**
Determine your savings in annual fuel expenses by switching from an SUV that averages 25 miles per gallon to a hybrid car averaging 50 miles per gallon.
**Calculation:**
1. **SUV Fuel Consumption:**
- 50,000 miles / 25 miles per gallon = 2,000 gallons
- 2,000 gallons × $4 per gallon = $8,000 per year
2. **Hybrid Car Fuel Consumption:**
- 50,000 miles / 50 miles per gallon = 1,000 gallons
- 1,000 gallons × $4 per gallon = $4,000 per year
3. **Annual Savings:**
- $8,000 - $4,000 = $4,000
**b. Annuity Savings Calculation:**
If you deposit your monthly fuel savings at the end of each month into an annuity that pays 4.8% interest compounded monthly, determine the total savings at the end of five years.
**Parameters:**
- Monthly savings: $4,000 / 12 = $333.33
- Interest rate (annually): 4.8% = 0.048
- Compounded monthly: \( n = 12 \)
- Total period: 5 years
**Using the formula:**
1. Plug values into the annuity formula:
\[ A = 333.33 \left[ \frac{\left(1 + \frac{0.048}{12}\right)^{12 \times 5} - 1}{\frac{0.048}{12}} \right] \]
2. Calculate to find the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6030cbbc-dba6-4cc4-b2c7-5d66af95ec0b%2F647e4223-295a-4583-b8ed-886776448e87%2Floa67ya_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Savings and Annuity Calculations**
**Formula for Annuity:**
\[ A = P \left[ \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}} \right] \]
Where:
- \( A \) = the future value of the annuity
- \( P \) = the payment amount per period
- \( r \) = annual interest rate
- \( n \) = number of times interest is compounded per year
- \( t \) = number of years
**Scenario:**
Suppose you drive 50,000 miles per year and gas averages $4 per gallon. Calculate the following:
**a. Annual Fuel Savings:**
Determine your savings in annual fuel expenses by switching from an SUV that averages 25 miles per gallon to a hybrid car averaging 50 miles per gallon.
**Calculation:**
1. **SUV Fuel Consumption:**
- 50,000 miles / 25 miles per gallon = 2,000 gallons
- 2,000 gallons × $4 per gallon = $8,000 per year
2. **Hybrid Car Fuel Consumption:**
- 50,000 miles / 50 miles per gallon = 1,000 gallons
- 1,000 gallons × $4 per gallon = $4,000 per year
3. **Annual Savings:**
- $8,000 - $4,000 = $4,000
**b. Annuity Savings Calculation:**
If you deposit your monthly fuel savings at the end of each month into an annuity that pays 4.8% interest compounded monthly, determine the total savings at the end of five years.
**Parameters:**
- Monthly savings: $4,000 / 12 = $333.33
- Interest rate (annually): 4.8% = 0.048
- Compounded monthly: \( n = 12 \)
- Total period: 5 years
**Using the formula:**
1. Plug values into the annuity formula:
\[ A = 333.33 \left[ \frac{\left(1 + \frac{0.048}{12}\right)^{12 \times 5} - 1}{\frac{0.048}{12}} \right] \]
2. Calculate to find the
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