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Andrei has a glass tank that he would like to fill with marbles and water.

W(m) models the amount of water Andrei needs (in liters) if he uses 
m marbles.





m
10
32
68



W(m)
36.5
28.8
16.2




When does the necessary amount of water decrease faster?
Choose 1 answer:
(A) Between 10 marbles and 32 marbles
(B) Between 32 marbles and 68 marbles
(C) The necessary amount of water decreases at the same rate over both intervals

Andrei has a glass tank that he would like to fill with marbles and water.\newlineW(m) W(m) models the amount of water Andrei needs (in liters) if he uses m m marbles.\newlinem m 1010 3232 6868 \newlineW(m) W(m) 3636.55 2828.88 1616.22\newline\end{tabular}\newlineWhen does the necessary amount of water decrease faster?\newlineChoose 11 answer:\newline(A) Between 1010 marbles and 3232 marbles\newline(B) Between 3232 marbles and 6868 marbles\newline(C) The necessary amount of water decreases at the same rate over both intervals

Full solution

Q. Andrei has a glass tank that he would like to fill with marbles and water.\newlineW(m) W(m) models the amount of water Andrei needs (in liters) if he uses m m marbles.\newlinem m 1010 3232 6868 \newlineW(m) W(m) 3636.55 2828.88 1616.22\newline\end{tabular}\newlineWhen does the necessary amount of water decrease faster?\newlineChoose 11 answer:\newline(A) Between 1010 marbles and 3232 marbles\newline(B) Between 3232 marbles and 6868 marbles\newline(C) The necessary amount of water decreases at the same rate over both intervals
  1. Calculate average rate of change: To determine when the necessary amount of water decreases faster, we need to calculate the average rate of change of the water function W(m)W(m) over the two intervals: [10,32][10, 32] and [32,68][32, 68].
  2. Calculate average rate of change between 1010 and 3232 marbles: First, we calculate the average rate of change of W(m)W(m) between 1010 marbles and 3232 marbles using the formula: (W(32)W(10))/(3210)(W(32) - W(10)) / (32 - 10).
  3. Substitute values into formula: Substitute the given values into the formula: (28.836.5)/(3210)=(7.7)/22=0.35(28.8 - 36.5) / (32 - 10) = (-7.7) / 22 = -0.35 (rounded to two decimal places).
  4. Calculate average rate of change between 3232 and 6868 marbles: Now, we calculate the average rate of change of W(m)W(m) between 3232 marbles and 6868 marbles using the formula: (W(68)W(32))/(6832)(W(68) - W(32)) / (68 - 32).
  5. Substitute values into formula: Substitute the given values into the formula: (16.228.8)/(6832)=(12.6)/36=0.35(16.2 - 28.8) / (68 - 32) = (-12.6) / 36 = -0.35 (rounded to two decimal places).
  6. Compare average rates of change: Comparing the two average rates of change, we see that they are both 0.35-0.35. This means that the necessary amount of water decreases at the same rate over both intervals.

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