Question

find the cost of each item in 15 years, assuming an annual inflation rate of 6% (compounded annually). (round your answers to the nearest cent.)
(a) movie admission, $10.00
$
(b) cd, $16.25
$
(c) textbook, $85.00
$
(d) electricity bill, $125.00
$

Explanation:

To find the cost of an item after \( t \) years with an annual inflation rate \( r \) compounded annually, we use the compound - interest formula for future value: \( A = P(1 + r)^t \), where:

• \( P \) is the present value (current cost) of the item.
• \( r \) is the annual inflation rate (expressed as a decimal).
• \( t \) is the number of years.
• \( A \) is the future value (cost after \( t \) years).

Here, \( r=0.06 \) (since \( 6\%=0.06 \)) and \( t = 15 \) years.

Part (a): Movie Admission (\( P=\$10.00 \))

Step 1: Identify the values

We have \( P = 10 \), \( r=0.06 \), and \( t = 15 \).

Step 2: Apply the formula

Substitute the values into the formula \( A=P(1 + r)^t \):
\( A=10\times(1 + 0.06)^{15} \)
First, calculate \( (1 + 0.06)^{15}\approx2.396558193 \)
Then, \( A = 10\times2.396558193\approx23.97 \)

Part (b): CD (\( P = \$16.25 \))

Step 1: Identify the values

We have \( P=16.25 \), \( r = 0.06 \), and \( t = 15 \).

Step 2: Apply the formula

Substitute into the formula \( A = P(1 + r)^t \):
\( A=16.25\times(1 + 0.06)^{15} \)
Since \( (1 + 0.06)^{15}\approx2.396558193 \)
\( A=16.25\times2.396558193\approx16.25\times2.3966\approx38.94 \) (rounded to the nearest cent)

Part (c): Textbook (\( P=\$85.00 \))

Step 1: Identify the values

We have \( P = 85 \), \( r=0.06 \), and \( t = 15 \).

Step 2: Apply the formula

Substitute into the formula \( A=P(1 + r)^t \):
\( A = 85\times(1 + 0.06)^{15} \)
Since \( (1 + 0.06)^{15}\approx2.396558193 \)
\( A=85\times2.396558193\approx85\times2.3966\approx203.71 \) (rounded to the nearest cent)

Part (d): Electricity Bill (\( P=\$125.00 \))

Step 1: Identify the values

We have \( P = 125 \), \( r=0.06 \), and \( t = 15 \).

Step 2: Apply the formula

Substitute into the formula \( A=P(1 + r)^t \):
\( A=125\times(1 + 0.06)^{15} \)
Since \( (1 + 0.06)^{15}\approx2.396558193 \)
\( A=125\times2.396558193\approx125\times2.3966 = 299.575\approx299.58 \) (rounded to the nearest cent)

Answer:

(a) \(\$23.97\)

(b) \(\$38.94\)

(c) \(\$203.71\)

(d) \(\$299.58\)