To determine how many model boats will be manufactured in the 12th week, we first need to understand the pattern of production.
The production starts with 62 units in the first week and increases by 75 units each subsequent week. The production for each week can be modeled by the formula for an arithmetic sequence:
\[ a_n = a_1 + (n - 1) \cdot d \]
Where:
- \(a_n\) is the number of units produced in week \(n\),
- \(a_1\) is the number of units produced in the first week (62 units),
- \(d\) is the common difference (75 units), and
- \(n\) is the week number.
For the 12th week (\(n = 12\)):
- \(a_1 = 62\)
- \(d = 75\)
Now we can calculate \(a_{12}\):
\[ a_{12} = 62 + (12 - 1) \cdot 75 \] \[ a_{12} = 62 + 11 \cdot 75 \] \[ a_{12} = 62 + 825 \] \[ a_{12} = 887 \]
Therefore, the number of model boats manufactured in the 12th week will be 887.