Questions: Find the cost of each item in 12 years, assuming an inflation rate of 11% (compounded continuously). (Round your answers to the nearest cent.) (a) phone bill, 44 164.71 (b) pair of shoes, 63 235.84 (c) new suit, 314 1175.43 (d) monthly rent, 750

Find the cost of each item in 12 years, assuming an inflation rate of 11% (compounded continuously). (Round your answers to the nearest cent.)
(a) phone bill, 44
164.71
(b) pair of shoes, 63
235.84
(c) new suit, 314
1175.43
(d) monthly rent, 750

Transcript text: Find the cost of each item in 12 years, assuming an inflation rate of $11 \%$ (compounded continuously). (Round your answers to the nearest cent.) (a) phone bill, $\$ 44$ \$ 164.71 (b) pair of shoes, $\$ 63$ \$ 235.84 (c) new suit, $\$ 314$ \$ 1175.43 (d) monthly rent, \$750 \$ $\square$

Solution

Solution Steps

Step 1: Convert the Annual Inflation Rate to a Decimal

To convert the annual inflation rate from a percentage to a decimal, we divide it by 100. Thus, $r = 11 / 100 = 0.11$.

Step 2: Use the Continuous Compounding Formula to Calculate Future Cost

The formula to calculate the future cost $F$ of an item with an initial cost $P$ is given by $F = P \cdot e^{rt}$, where $e$ is the base of the natural logarithm, $r$ is the annual inflation rate expressed as a decimal, and $t$ is the time in years. Substituting the given values, we get $F = 750 \cdot e^{0.11 \cdot 12}$.

Step 3: Round the Result to the Nearest Cent

After calculating the future cost using the formula, we round the result to the nearest cent (or the specified number of decimal places). Thus, the future cost rounded to 2 decimal places is $F = 2807.57$.