Question

A TV manufacturer produces 500 TVs in its first year and plans to increase production by 8 percent each year for the next 10 years. How many total TVs will have been produced at the end of the tenth year? (1 point) Responses 6,856 6,856 8,949 8,949 7,243 7,243 5,687

Answer 1

To calculate the total number of TVs produced at the end of the tenth year, we need to find the production for each year, given that the production increases by 8 percent each year.

The production for each year can be represented as:
- Year 1: \( P_1 = 500 \)
- Year 2: \( P_2 = P_1 \times 1.08 = 500 \times 1.08 \)
- Year 3: \( P_3 = P_2 \times 1.08 = 500 \times (1.08)^2 \)
- Year 4: \( P_4 = P_3 \times 1.08 = 500 \times (1.08)^3 \)
- ...
- Year 10: \( P_{10} = P_1 \times (1.08)^9 \)

The total production over 10 years \( T \) can be calculated using the formula for the sum of a geometric series. The total number of TVs produced in the 10 years can be calculated using the sum:
\[
T = P_1 \frac{(1 + r + r^2 + \ldots + r^{n-1})}{n}
\]
where:
- \( P_1 = 500 \) (initial production)
- \( r = 1.08 \) (growth rate)
- \( n = 10 \) (number of years)

The formula for the sum of a geometric series can also be expressed as:
\[
T = P_1 \frac{(1 - r^n)}{(1 - r)}
\]

Plugging in the values:
\[
T = 500 \frac{(1 - (1.08)^{10})}{(1 - 1.08)}
\]

Calculating \( (1.08)^{10} \):
\[
(1.08)^{10} \approx 2.15892
\]

Now plugging it back in:
\[
T = 500 \frac{(1 - 2.15892)}{-0.08}
\]

Calculating \( (1 - 2.15892) \):
\[
1 - 2.15892 \approx -1.15892
\]

Now substituting:
\[
T = 500 \frac{-1.15892}{-0.08}
\]
\[
T = 500 \times 14.4865
\]
\[
T \approx 7243.25
\]

Since we cannot produce a fraction of a TV, we round down to the nearest whole number:
\[
T \approx 7243
\]

Thus, the total number of TVs produced at the end of the tenth year is approximately 7243.

Therefore, the answer is **7,243**.