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let ( s(t) ) give the number of acres of wetlands in a state in year ( t ). assume that the area of wetlands goes down over time. what does the statement ( \frac{s(33) - s(13)}{33 - 13} = -506 ) tell you about the wetlands?

○ this tells us that between year ( t = 13 ) and ( t = 33 ), the area of wetlands increased by 506 acres.
○ this tells us that between year ( t = 13 ) and ( t = 33 ), the area of wetlands was 506 acres.
○ this tells us that between year ( t = 13 ) and ( t = 33 ), the area of wetlands increased at an average rate of 506 acres per year.
○ this tells us that between year ( t = 13 ) and ( t = 33 ), the area of wetlands decreased at an average rate of 506 acres per year.
○ this tells us that between year ( t = 13 ) and ( t = 33 ), the area of wetlands decreased by 506 acres.

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Explanation:

1. Recall the formula for the average rate of change of a function \( s(t) \) over the interval \([a,b]\) is \(\frac{s(b)-s(a)}{b - a}\). Here, \( a = 13 \), \( b=33 \), so the formula \(\frac{s(33)-s(13)}{33 - 13}\) represents the average rate of change of the wetland area with respect to time (in years) between \( t = 13 \) and \( t=33 \).
2. The value of this average rate of change is \(- 506\). The negative sign indicates a decrease (since the area of wetlands goes down over time), and the magnitude \( 506 \) represents the number of acres per year. So it means the area of wetlands decreased at an average rate of 506 acres per year between \( t = 13 \) and \( t = 33 \).

• Option A is wrong because the negative sign implies a decrease, not an increase.
• Option B is wrong as the formula does not give the area of wetlands at a particular time or the total area, but the rate of change.
• Option C is wrong because the negative sign shows a decrease, not an increase in the rate.
• Option E is wrong because the formula gives the rate of change (acres per year), not the total change in area (the total change would be \( s(33)-s(13)=- 506\times(33 - 13)=-506\times20=- 10120\) acres, not - 506 acres).

Answer:

D. This tells us that between year \( t = 13 \) and \( t = 33 \), the area of wetlands decreased at an average rate of 506 acres per year.