Question

the graph shows the amount of water in a water tank over several hours.
amount of water in cistern
water (thousands of gallons)
0 2 4 6 8 10
hours
12
10
8
6
4
2
0
a. what is the average rate of change on the interval 0,4 and on the interval 6,10? what is a possible explanation for what each rate of change indicates?
a. what is the average rate of change on the interval 0,4?

gallons per hour

Explanation:

Step1: Identify points at x=0 and x=4

At \( x = 0 \), the water amount \( y_1 = 12 \) (thousand gallons). At \( x = 4 \), \( y_2 = 4 \) (thousand gallons).

Step2: Apply average rate of change formula

The formula for average rate of change on \([a,b]\) is \( \frac{y_2 - y_1}{b - a} \). Here, \( a = 0 \), \( b = 4 \), \( y_1 = 12 \), \( y_2 = 4 \). So, \( \frac{4 - 12}{4 - 0} = \frac{-8}{4} = -2 \) (thousand gallons per hour, but since the question says "gallons per hour", we note the graph is in thousands, so -2000? Wait, no, the y-axis is "thousands of gallons". Wait, the question's box is for gallons per hour. Wait, the graph: at x=0, y=12 (thousand gallons) = 12000 gallons. At x=4, y=4 (thousand gallons) = 4000 gallons. So average rate of change is \( \frac{4000 - 12000}{4 - 0} = \frac{-8000}{4} = -2000 \) gallons per hour? Wait, no, maybe the y-axis is in thousands, but the question says "gallons per hour". Wait, the graph's y-axis label is "Water (thousands of gallons)". So at x=0, it's 12 thousand gallons = 12,000 gallons. At x=4, it's 4 thousand gallons = 4,000 gallons. So the change in y is 4000 - 12000 = -8000 gallons. Change in x is 4 - 0 = 4 hours. So average rate of change is \( \frac{-8000}{4} = -2000 \) gallons per hour? Wait, but maybe the y-axis is in thousands, so the values are 12, 10, 8, etc., in thousands. So 12 thousand gallons at x=0, 4 thousand at x=4. So the formula is \( \frac{4 - 12}{4 - 0} \) thousand gallons per hour, which is -2 thousand gallons per hour, i.e., -2000 gallons per hour. But let's check the graph again. The first point is (0,12), the point at x=4 is (4,4). So using the average rate of change formula: \( \text{Average rate of change} = \frac{f(4) - f(0)}{4 - 0} \). \( f(0) = 12 \) (thousand gallons), \( f(4) = 4 \) (thousand gallons). So \( \frac{4 - 12}{4} = \frac{-8}{4} = -2 \) thousand gallons per hour, which is -2000 gallons per hour. Wait, but maybe the question considers the y-axis as gallons, not thousands? But the label is "thousands of gallons". Hmm. Wait, maybe the graph is in thousands, but the question asks for gallons per hour. So 12 thousand gallons is 12,000 gallons, 4 thousand is 4,000. So the change is 4000 - 12000 = -8000 gallons over 4 hours, so -2000 gallons per hour. But let's confirm the formula. The average rate of change on [a,b] is \( \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \). So x1=0, y1=12 (thousand gallons); x2=4, y2=4 (thousand gallons). So \( \frac{4 - 12}{4 - 0} = \frac{-8}{4} = -2 \) thousand gallons per hour, which is -2000 gallons per hour. So the answer is -2000? Wait, but maybe the y-axis is in gallons, not thousands. Let me check the graph again. The y-axis has 0, 2, 4, 6, 8, 10, 12. If it's in gallons, 12 gallons at x=0, 4 at x=4. Then the average rate of change is (4 - 12)/(4 - 0) = -2 gallons per hour. But that seems too low. The graph is a line from (0,12) to (4,4), then flat to (6,4), then up to (10,10). So if y is in thousands, then 12,000 to 4,000 over 4 hours: 8,000 gallons lost in 4 hours, so 2,000 per hour. So the average rate of change is -2 (thousand gallons per hour) or -2000 gallons per hour. But the question's box is for "gallons per hour". So probably, the y-axis is in thousands, so we need to convert. Wait, maybe the problem has a typo, or the y-axis is in gallons. Let's assume the y-axis is in gallons. Then (0,12) and (4,4). So average rate of change is (4 - 12)/(4 - 0) = -2 gallons per hour. But that seems odd. Wait, the graph's first point is (0,12), then at x=2, it's 8, x=4, 4. So the slope from 0 to 4 is (4-12)/(4-0) = -2. So if y is in gallons, then -2 gallons per hour. But that seems too slow. Alternatively, if y is in thousands, then -2 thousand gallons per hour, i.e., -2000 gallons per hour. Let's check the problem statement again: "the amount of water in a water tank over several hours". A water tank with 12 gallons? No, that's too small. So it's 12 thousand gallons. So the average rate of change is -2 thousand gallons per hour, which is -2000 gallons per hour. So the answer is -2000? Wait, but maybe the question expects the answer in thousands, but the unit is gallons per hour. Wait, no, the unit is "gallons per hour". So 12 thousand gallons is 12,000 gallons. So 12,000 to 4,000 over 4 hours: (4,000 - 12,000)/4 = -8,000/4 = -2,000 gallons per hour. So the average rate of change is -2000 gallons per hour.

Answer:

-2000