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 8,949 8,949 7,243 7,243 5,687 5,687 6,856
Answers
GPT-4o mini
To calculate the total number of TVs produced over the 10 years with an initial production of 500 TVs and an annual increase of 8%, we can use the formula for the sum of a geometric series, as the production forms a geometric progression.
The formula for the number of TVs produced in year \( n \) is given by:
\[
P_n = P_0 \cdot (1 + r)^{n-1}
\]
where:
- \( P_n \) is the number of TVs produced in year \( n \),
- \( P_0 \) is the initial production (500 TVs),
- \( r \) is the growth rate (8% or 0.08),
- \( n \) is the year number.
We want to calculate the total production for 10 years:
\[
\text{Total Production} = \sum_{n=1}^{10} P_n = P_0 \cdot \sum_{n=0}^{9} (1 + r)^n
\]
The sum of the first \( N \) terms of a geometric series can be calculated using:
\[
S_N = a \cdot \frac{1 - r^N}{1 - r}
\]
where \( a \) is the first term and \( r \) is the common ratio.
Here,
- \( a = 500 \)
- \( r = 1.08 \)
- \( N = 10 \)
First, calculate the sum of the first 10 terms:
\[
\sum_{n=0}^{9} (1.08)^n = \frac{1 - (1.08)^{10}}{1 - 1.08}
\]
Calculating \( (1.08)^{10} \):
\[
(1.08)^{10} \approx 2.1589
\]
Now plug into the sum formula:
\[
\sum_{n=0}^{9} (1.08)^n = \frac{1 - 2.1589}{1 - 1.08} = \frac{-1.1589}{-0.08} \approx 14.48625
\]
Now multiplying this by the initial production:
\[
\text{Total Production} = 500 \cdot 14.48625 \approx 7243.13
\]
Rounding to the nearest whole number, the total number of TVs produced at the end of the tenth year is **7243**.
Thus, the answer is **7,243**.
The formula for the number of TVs produced in year \( n \) is given by:
\[
P_n = P_0 \cdot (1 + r)^{n-1}
\]
where:
- \( P_n \) is the number of TVs produced in year \( n \),
- \( P_0 \) is the initial production (500 TVs),
- \( r \) is the growth rate (8% or 0.08),
- \( n \) is the year number.
We want to calculate the total production for 10 years:
\[
\text{Total Production} = \sum_{n=1}^{10} P_n = P_0 \cdot \sum_{n=0}^{9} (1 + r)^n
\]
The sum of the first \( N \) terms of a geometric series can be calculated using:
\[
S_N = a \cdot \frac{1 - r^N}{1 - r}
\]
where \( a \) is the first term and \( r \) is the common ratio.
Here,
- \( a = 500 \)
- \( r = 1.08 \)
- \( N = 10 \)
First, calculate the sum of the first 10 terms:
\[
\sum_{n=0}^{9} (1.08)^n = \frac{1 - (1.08)^{10}}{1 - 1.08}
\]
Calculating \( (1.08)^{10} \):
\[
(1.08)^{10} \approx 2.1589
\]
Now plug into the sum formula:
\[
\sum_{n=0}^{9} (1.08)^n = \frac{1 - 2.1589}{1 - 1.08} = \frac{-1.1589}{-0.08} \approx 14.48625
\]
Now multiplying this by the initial production:
\[
\text{Total Production} = 500 \cdot 14.48625 \approx 7243.13
\]
Rounding to the nearest whole number, the total number of TVs produced at the end of the tenth year is **7243**.
Thus, the answer is **7,243**.