4. The company AMBICIOZNI has made a production plan for 2011 and the next few years. They plan for production to double every subsequent year. Production in the final year of the planning period will be 320 tons, and the total production over these years will amount to 630 tons. What will the production be in 2011? How many years ahead are they planning?

(6 points)
4. Let $\mathrm{Z} a_{1}$ denote the production quantity in 2010 and $\mathrm{z} a_{n}$ the production quantity in the last planned year, and $\mathrm{s} s_{n}$ the total planned production quantity over the planned period. Since the production doubles every subsequent year, $q=2$. Therefore, $a_{n}=a_{1} \cdot q^{n-1}$, so $320=a_{1} \cdot 2^{n-1}$. It also holds that $s_{n}=\frac{a_{1}\left(2^{n}-1\right)}{2-1}$, so $630=a_{1} \cdot\left(2^{n}-1\right)$. From the first equation, we express $a_{1}=\frac{320}{2^{n-1}}$ and substitute it into the second equation $630=\frac{320}{2^{n-1}} \cdot\left(2^{n}-1\right)$. Eliminate the fraction $2^{n-1} \cdot 630=320 \cdot\left(2^{n}-1\right)$, rearrange the equation $630 \cdot 2^{n-1}-320 \cdot 2^{n}=-320$. Factor out the common term $2^{n-1}(320 \cdot 2-630)=320$, rearrange $2^{n-1} \cdot 10=320$, divide by 10. We get $2^{n-1}=32$, which can be written as $2^{n-1}=2^{5}$. Calculate $n=6$. Then calculate $a_{1}=10$. The production quantity in 2010 is $a_{1}=10$ tons. The production plan will be realized in 6 years.

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-12.jpg?height=48&width=1639&top_left_y=1295&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-12.jpg?height=51&width=1639&top_left_y=1345&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-12.jpg?height=68&width=1639&top_left_y=1391&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-12.jpg?height=52&width=1636&top_left_y=1453&top_left_x=210)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-12.jpg?height=49&width=1642&top_left_y=1506&top_left_x=207)

Answer: The production in 2010 is 10 tons, the plan will be realized in 6 years....... 1 point

NOTE: A half-correct answer scores 0 points.