Carmen and Matt are conducting different chemistry experiments in school. In both experiments, each student starts with an initi
al amount of water in a flask. They combine two chemicals which react to produce more water. Carmen's experiment starts with 30 milliliters of water in a flask, and the water increases in volume by 8.5 milliliters per second. Matt's experiment starts with 10 milliliters of water and increases in volume by 28% each second. The graph represents the volume of water in the two flasks in relation to time. Which two conclusions can be made if f represents the volume of water in Carmen's flask and g represents the volume of water in Matt's flask? The volume of water in Carmen's flask is increasing at a slower rate than the volume of water in Matt's flask over the interval [0, 2]. The volume of water in Carmen's flask is increasing at a faster rate than the volume of water in Matt's flask over the interval [6, 8]. The volume of water in Carmen's flask will always be greater than the volume of water in Matt's flask. The volume of water in Matt's flask will eventually be greater than the volume of water in Carmen's flask. The volume of water in Carmen's flask is increasing at a slower rate than the volume of water in Matt's flask over the interval [4, 6].
2 answers:
Answer:
- The volume of water in Matt's flask will eventually be greater than the volume of water in Carmen's flask.
- The volume of water in Carmen's flask is increasing at a slower rate than the volume of water in Matt's flask over the interval [4, 6].
Step-by-step explanation:
The attached graph shows the results from the experiments. Rays are drawn showing the average rate of change for the intervals [0,2], [4,6], and [6,8].
The average rate of change of volume in Matt's flask over the interval [4, 6] is very nearly the same as Carmen's. However, Carmen's is slightly slower.
Here's the average rate of change in Matt's flask on that interval:
(g(6) -g(4))/(6 -4) ≈ (43.980 -26.844)/2 = 17.136/2 = 8.568 . . . mL/s
This is very slightly more than Carmen's rate of 8.5 mL/s.
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An exponential function will eventually exceed any polynomial function, so Matt's result will eventually exceed Carmen's. (It does so before 10 seconds pass.)
_____
On the interval [0, 2], Carmen's rate of increase clearly exceeds Matt's. On the interval [6, 8], the reverse is true. The volume in Carmen's flask is only greater than Matt's for a short period (less than about 9.8 seconds).
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