Answer :
To determine the function that represents the number of windmills, [tex]\( w \)[/tex], on the windfarm [tex]\( t \)[/tex] years after it opens, let's take a step-by-step approach:
1. Initial Number of Windmills:
The number of windmills at the windfarm starts at [tex]\( 200 \)[/tex]. This is the initial count.
2. Windmills Added Each Year:
Each year, [tex]\( 6 \)[/tex] new windmills are added to the windfarm.
3. Windmills Removed Each Year:
Each year, [tex]\( 2 \)[/tex] windmills are removed due to wear and tear.
4. Net Change in Windmills Each Year:
To find the net change in the number of windmills per year, we subtract the number of windmills removed from the number of windmills added:
[tex]\[ \text{Net change per year} = \text{Windmills added per year} - \text{Windmills removed per year} \][/tex]
Substituting the given values:
[tex]\[ \text{Net change per year} = 6 - 2 = 4 \][/tex]
5. Function for the Number of Windmills over Time:
To find the total number of windmills [tex]\( t \)[/tex] years after the windfarm opens, we start with the initial number of windmills and add the net change per year multiplied by the number of years [tex]\( t \)[/tex].
Therefore, the function [tex]\( w(t) \)[/tex] that represents the number of windmills is:
[tex]\[ w(t) = \text{Initial windmills} + \text{Net change per year} \times t \][/tex]
Substituting the values we have:
[tex]\[ w(t) = 200 + 4t \][/tex]
6. Conclusion:
The correct function that represents the number of windmills [tex]\( t \)[/tex] years after the windfarm opens is:
[tex]\[ \boxed{w(t) = 200 + 4t} \][/tex]
Thus, the correct answer is [tex]\( \text{(B)} \)[/tex].
1. Initial Number of Windmills:
The number of windmills at the windfarm starts at [tex]\( 200 \)[/tex]. This is the initial count.
2. Windmills Added Each Year:
Each year, [tex]\( 6 \)[/tex] new windmills are added to the windfarm.
3. Windmills Removed Each Year:
Each year, [tex]\( 2 \)[/tex] windmills are removed due to wear and tear.
4. Net Change in Windmills Each Year:
To find the net change in the number of windmills per year, we subtract the number of windmills removed from the number of windmills added:
[tex]\[ \text{Net change per year} = \text{Windmills added per year} - \text{Windmills removed per year} \][/tex]
Substituting the given values:
[tex]\[ \text{Net change per year} = 6 - 2 = 4 \][/tex]
5. Function for the Number of Windmills over Time:
To find the total number of windmills [tex]\( t \)[/tex] years after the windfarm opens, we start with the initial number of windmills and add the net change per year multiplied by the number of years [tex]\( t \)[/tex].
Therefore, the function [tex]\( w(t) \)[/tex] that represents the number of windmills is:
[tex]\[ w(t) = \text{Initial windmills} + \text{Net change per year} \times t \][/tex]
Substituting the values we have:
[tex]\[ w(t) = 200 + 4t \][/tex]
6. Conclusion:
The correct function that represents the number of windmills [tex]\( t \)[/tex] years after the windfarm opens is:
[tex]\[ \boxed{w(t) = 200 + 4t} \][/tex]
Thus, the correct answer is [tex]\( \text{(B)} \)[/tex].
