Question: A company makes two kinds of products: Basic and Advanced. To manufacture the Basic takes $2$ hours in Assembly and $2.75$ hours in Finishing. To manufacture the Advanced takes $3.5$ hours in Assembly and $2.15$ hours in Finishing. The Assembly department has $2645$ hours per week and the Finishing department has $2083$ hours per week. How many of each type of product should be manufactured to run at capacity?
If necessary, round down to the nearest whole number.
The number of Basic units should be [blank] .
The number of Advanced units should be [blank] .
1. **State the problem:**
We need to find how many Basic ($x$) and Advanced ($y$) products to manufacture so that the Assembly and Finishing departments run at full capacity.
2. **Set up the system of equations:**
- Assembly time: $2x + 3.5y = 2645$
- Finishing time: $2.75x + 2.15y = 2083$
3. **Solve the system:**
From the first equation:
$$2x + 3.5y = 2645$$
Multiply the second equation by $2$ and the first by $2.75$ to align coefficients for elimination:
$$2.75(2x + 3.5y) = 2.75 \times 2645 \Rightarrow 5.5x + 9.625y = 7273.75$$
$$2(2.75x + 2.15y) = 2 \times 2083 \Rightarrow 5.5x + 4.3y = 4166$$
Subtract the second from the first:
$$ (5.5x + 9.625y) - (5.5x + 4.3y) = 7273.75 - 4166 $$
$$ 5.325y = 3107.75 $$
$$ y = \frac{3107.75}{5.325} \approx 583.68 $$
4. **Find $x$ using $y$ in the first equation:**
$$ 2x + 3.5(583.68) = 2645 $$
$$ 2x + 2042.88 = 2645 $$
$$ 2x = 2645 - 2042.88 = 602.12 $$
$$ x = \frac{602.12}{2} = 301.06 $$
5. **Round down to nearest whole number:**
$$ x = 301, \quad y = 583 $$
**Final answer:**
- Number of Basic units: $301$
- Number of Advanced units: $583$