To determine how many model boats will be manufactured in the 12th week, we notice that the production is described as an arithmetic sequence where each term increases by a constant amount (75 units) each week.
The sequence starts at 62 units in the first week, and the production increases by 75 units each subsequent week.
The first few terms of the sequence are:
- Week 1: 62
- Week 2: 62 + 75 = 137
- Week 3: 137 + 75 = 212
- Week 4: 212 + 75 = 287
To find the number of boats produced in the 12th week, we can use the formula for the nth term of an arithmetic sequence:
\[ a_n = a_1 + (n - 1) \cdot d \]
where:
- \( a_n \) is the \( n \)-th term.
- \( a_1 \) is the first term (62).
- \( n \) is the week number (12).
- \( d \) is the common difference (75).
Substituting in the values:
\[ a_{12} = 62 + (12 - 1) \cdot 75 \] \[ a_{12} = 62 + 11 \cdot 75 \] \[ a_{12} = 62 + 825 \] \[ a_{12} = 887 \]
Therefore, in the 12th week, 887 model boats will be manufactured.