which statement best describes the average rate of change in ( c(x) ) on the interval from ( x = 10 ) to ( x = 20 )\
$ \circ $ the cost per square foot decreases by 10 cents on average for each extra foot of length.\
$ \circ $ the cost per square foot is 10 cents less for a 20-foot-long enclosure than a 10-foot-long enclosure.\
$ \circ $ the cost per square foot decreases by 20 cents on average for each extra foot of length.\
$ \circ $ the cost per square foot is 20 cents less for a 20-foot-long enclosure than a 10-foot-long enclosure.

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Accepted Answer

# Brief Explanations:
To determine the correct statement about the average rate of change from $ x = 10 $ to $ x = 20 $ for the cost per square foot $ c(x) $, we analyze the average rate of change formula: $ \text{Average Rate of Change} = \frac{c(20) - c(10)}{20 - 10}=\frac{c(20)-c(10)}{10} $. The third option states "THE COST PER SQUARE FOOT IS 20 CENTS LESS FOR A 20 - FOOT - LONG ENCLOSURE THAN A 10 - FOOT - LONG ENCLOSURE". If $ c(20)=c(10)-0.20 $ (since 20 cents is $ 0.20 $ dollars), then $ \frac{c(20)-c(10)}{10}=\frac{(c(10)-0.20)-c(10)}{10}=\frac{- 0.20}{10}=-0.02 $ dollars per foot (or - 2 cents per foot). But wait, let's re - evaluate. Wait, the fourth option? Wait, no, let's check the options again. Wait, the option "THE COST PER SQUARE FOOT IS 20 CENTS LESS FOR A 20 - FOOT - LONG ENCLOSURE THAN A 10 - FOOT - LONG ENCLOSURE" means $ c(20)=c(10)-0.20 $. Then the average rate of change is $ \frac{c(20)-c(10)}{20 - 10}=\frac{(c(10)-0.20)-c(10)}{10}=\frac{-0.20}{10}=-0.02 $ dollars per foot ( - 2 cents per foot)? No, wait, maybe I misread. Wait, the option "THE COST PER SQUARE FOOT IS 20 CENTS LESS FOR A 20 - FOOT - LONG ENCLOSURE THAN A 10 - FOOT - LONG ENCLOSURE" is equivalent to $ c(20)=c(10)-0.20 $. Then the change in cost is $ c(20)-c(10)=- 0.20 $ dollars over a change in length of $ 20 - 10 = 10 $ feet. So the average rate of change is $ \frac{-0.20}{10}=-0.02 $ dollars per foot (or - 2 cents per foot). But the option "THE COST PER SQUARE FOOT IS 20 CENTS LESS FOR A 20 - FOOT - LONG ENCLOSURE THAN A 10 - FOOT - LONG ENCLOSURE" is the one that correctly describes the relationship between $ c(20) $ and $ c(10) $ in terms of the difference, which is related to the average rate of change. Wait, maybe my initial calculation was wrong. Let's assume that the cost per square foot for 10 - foot is $ c(10) $ and for 20 - foot is $ c(20) $. If the cost for 20 - foot is 20 cents less than for 10 - foot, then $ c(20)=c(10)-0.20 $. Then the average rate of change from 10 to 20 is $ \frac{c(20)-c(10)}{20 - 10}=\frac{(c(10)-0.20)-c(10)}{10}=\frac{-0.20}{10}=-0.02 $ dollars per foot ( - 2 cents per foot). But the option "THE COST PER SQUARE FOOT IS 20 CENTS LESS FOR A 20 - FOOT - LONG ENCLOSURE THAN A 10 - FOOT - LONG ENCLOSURE" is the one that directly states the difference between $ c(20) $ and $ c(10) $ as a 20 - cent decrease, which is related to the average rate of change over the interval from 10 to 20. The other options: the first option says a 10 - cent decrease per extra foot, the second says 10 cents less for 20 - foot than 19 - foot (not relevant to 10 - 20), the fourth option (wait, the third option? Wait, the options are:

1. THE COST PER SQUARE FOOT DECREASES BY 10 CENTS ON AVERAGE FOR EACH EXTRA FOOT OF LENGTH.

2. THE COST PER SQUARE FOOT IS 10 CENTS LESS FOR A 20 - FOOT - LONG ENCLOSURE THAN A 19 - FOOT - LONG ENCLOSURE.

3. THE COST PER SQUARE FOOT IS 20 CENTS LESS FOR A 20 - FOOT - LONG ENCLOSURE THAN A 10 - FOOT - LONG ENCLOSURE.

4. (Wait, the original image has four options? Wait, the user's image shows four options. Let's re - express:

Option 1: THE COST PER SQUARE FOOT DECREASES BY 10 CENTS ON AVERAGE FOR EACH EXTRA FOOT OF LENGTH.

Option 2: THE COST PER SQUARE FOOT IS 10 CENTS LESS FOR A 20 - FOOT - LONG ENCLOSURE THAN A 19 - FOOT - LONG ENCLOSURE.

Option 3: THE COST PER SQUARE FOOT DECREASES BY 20 CENTS ON AVERAGE FOR EACH EXTRA FOOT OF LENGTH.

Option 4: THE COST PER SQUARE FOOT IS 20 CENTS LESS FOR A 20 - FOOT - LONG ENCLOSURE THAN A 10 - FOOT - LONG ENCLOSURE.

Ah, I misread earlier. So option 4: "THE COST PER SQUARE FOOT IS 20 CENTS LESS FOR A 20 - FOOT - LONG ENCLOSURE THAN A 10 - FOOT - LONG ENCLOSURE" means $ c(20)=c(10)-0.20 $. Then the average rate of change from $ x = 10 $ to $ x = 20 $ is $ \frac{c(20)-c(10)}{20 - 10}=\frac{(c(10)-0.20)-c(10)}{10}=\frac{-0.20}{10}=-0.02 $ dollars per foot ( - 2 cents per foot). But option 4 is the one that correctly states the difference between $ c(20) $ and $ c(10) $ as a 20 - cent decrease, which is the total change over the interval from 10 to 20 (a length change of 10 feet). The other options: option 1 is about per - foot decrease (average rate), but if the total change over 10 feet is - 20 cents, then the average rate is - 2 cents per foot, not - 10. Option 2 is about 20 - foot vs 19 - foot, not 10 - 20. Option 3 is a per - foot decrease of 20 cents, which would mean total change of $ 20\times10 = 200 $ cents over 10 feet, which is not the case. So the correct option is the fourth one: "THE COST PER SQUARE FOOT IS 20 CENTS LESS FOR A 20 - FOOT - LONG ENCLOSURE THAN A 10 - FOOT - LONG ENCLOSURE".

# Answer:
THE COST PER SQUARE FOOT IS 20 CENTS LESS FOR A 20 - FOOT - LONG ENCLOSURE THAN A 10 - FOOT - LONG ENCLOSURE.

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