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Calculate the cost of each item in 13 years, assuming an annual inflation rate of 11% (compounded annually). Round your answers to the nearest cent.

a) Phone bill: $40

b) Pair of shoes: $65

c) New suit: $350

d) Monthly rent: $625

Sagot :

GinnyAnswer
To determine the cost of each of the given items in 13 years, assuming an annual inflation rate of 11%, we can use the compound interest formula:
[tex]\[ \text{Future Value} = \text{Present Value} \times (1 + r)^t \][/tex]
where:
- \(\text{Future Value}\) is the amount we are looking to find.
- \(\text{Present Value}\) is the current cost of the item.
- \(r\) is the inflation rate expressed as a decimal (11% = 0.11).
- \(t\) is the number of years (13 years).

Let's go through the calculation step-by-step for each item:

1. Phone Bill:
- Present Value (PV) = $40
- Inflation rate (r) = 0.11
- Number of years (t) = 13

Applying the formula:
[tex]\[ \text{Future Value} = 40 \times (1 + 0.11)^{13} \][/tex]
[tex]\[ \text{Future Value} \approx 40 \times 3.883225 \][/tex]
[tex]\[ \text{Future Value} \approx 155.33 \][/tex]
Therefore, the future cost of the phone bill will be approximately $155.33.

2. Pair of Shoes:
- Present Value (PV) = $65
- Inflation rate (r) = 0.11
- Number of years (t) = 13

Applying the formula:
[tex]\[ \text{Future Value} = 65 \times (1 + 0.11)^{13} \][/tex]
[tex]\[ \text{Future Value} \approx 65 \times 3.883225 \][/tex]
[tex]\[ \text{Future Value} \approx 252.41 \][/tex]
Therefore, the future cost of the pair of shoes will be approximately $252.41.

3. New Suit:
- Present Value (PV) = $350
- Inflation rate (r) = 0.11
- Number of years (t) = 13

Applying the formula:
[tex]\[ \text{Future Value} = 350 \times (1 + 0.11)^{13} \][/tex]
[tex]\[ \text{Future Value} \approx 350 \times 3.883225 \][/tex]
[tex]\[ \text{Future Value} \approx 1359.15 \][/tex]
Therefore, the future cost of the new suit will be approximately $1359.15.

4. Monthly Rent:
- Present Value (PV) = $625
- Inflation rate (r) = 0.11
- Number of years (t) = 13

Applying the formula:
[tex]\[ \text{Future Value} = 625 \times (1 + 0.11)^{13} \][/tex]
[tex]\[ \text{Future Value} \approx 625 \times 3.883225 \][/tex]
[tex]\[ \text{Future Value} \approx 2427.05 \][/tex]
Therefore, the future cost of the monthly rent will be approximately $2427.05.

In summary, the costs in 13 years will be:
- Phone bill: $155.33
- Pair of shoes: $252.41
- New Suit: $1359.15
- Monthly Rent: $2427.05
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