To determine whether the family will have enough money for their vacation, let's calculate their total savings for the months of March through August.
- Initial Savings: $125
- Expected Cost of Vacation: $1364
- Monthly Deposits: They will deposit 20% more each month than the previous month.
Let's calculate the deposits for each month:
- Month 1 (March):
- Deposit = \( x_1 \)
- Month 2 (April):
- Deposit = \( x_2 = x_1 \times 1.2 \)
- Month 3 (May):
- Deposit = \( x_3 = x_2 \times 1.2 = x_1 \times 1.2^2 \)
- Month 4 (June):
- Deposit = \( x_4 = x_3 \times 1.2 = x_1 \times 1.2^3 \)
- Month 5 (July):
- Deposit = \( x_5 = x_4 \times 1.2 = x_1 \times 1.2^4 \)
- Month 6 (August):
- Deposit = \( x_6 = x_5 \times 1.2 = x_1 \times 1.2^5 \)
We will also need to recognize that they can choose an initial deposit amount. If we denote the initial deposit in March as \( x_1 \), then their deposits for these six months will be:
- March: \( x_1 \)
- April: \( x_1 \times 1.2 \)
- May: \( x_1 \times 1.2^2 \)
- June: \( x_1 \times 1.2^3 \)
- July: \( x_1 \times 1.2^4 \)
- August: \( x_1 \times 1.2^5 \)
The total amount saved by the end of August becomes:
\[ \text{Total Savings} = 125 + x_1 + x_1(1.2) + x_1(1.2^2) + x_1(1.2^3) + x_1(1.2^4) + x_1(1.2^5) \]
This can be simplified using the formula for the sum of a geometric series, \( S_n = a \frac{(1-r^n)}{(1-r)} \) where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.
Here:
- \( a = x_1 \)
- \( r = 1.2 \)
- \( n = 6 \)
The sum of deposits made will then be:
\[ S = x_1 \frac{(1 - 1.2^6)}{(1 - 1.2)} = x_1 \frac{(1 - 2.985984)}{-0.2} \approx x_1 \cdot 9.92992 \]
Thus, adding the initial $125 to the sum of the deposits gives:
\[ \text{Total Savings} = 125 + 9.92992 x_1 \]
To ensure they have enough, this total must be at least $1364:
\[ 125 + 9.92992 x_1 \geq 1364 \]
Isolating \( x_1 \):
\[ 9.92992 x_1 \geq 1364 - 125 \] \[ 9.92992 x_1 \geq 1239 \] \[ x_1 \geq \frac{1239}{9.92992} \approx 124.569 \]
Thus, to know how much money they need for their deposits starting at March, they would need to start depositing at least:
\[ x_1 \approx 125.569 \text{ (rounded up to } 126\text{)} \]
Now, let's check how much they would actually have if they started with the minimum required (126):
After checking with \( x_1 = 126 \):
- March: 126
- April: 151.2
- May: 181.44
- June: 217.728
- July: 261.2736
- August: 313.52832
Total =
\[ 125 + 126 + 151.2 + 181.44 + 217.728 + 261.2736 + 313.52832 \approx 1375.60192 \]
This means they would have:
\( 1375.60192 - 1364 = 11.60192 \)
Thus, they can actually afford their trip with \( x_1 \approx 126 \).
Conclusion
Answer B: Yes, they will have enough money.
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