# A scuba diver dove from the surface of the ocean to an elevation of −79frac{9}{10} feet at a rate of - 18.8 feet per minute. After spending 12.75 minutes at that elevation, the diver ascended to an elevation of −28frac{9}{10} feet. The total time for the dive so far was 19frac{1}{8}. minutes. What was the rate of change in the diver's elevation during the ascent?

Question
Factors and multiples
A scuba diver dove from the surface of the ocean to an elevation of $$\displaystyle−{79}{\frac{{{9}}}{{{10}}}}$$ feet at a rate of - 18.8 feet per minute. After spending 12.75 minutes at that elevation, the diver ascended to an elevation of $$\displaystyle−{28}{\frac{{{9}}}{{{10}}}}$$ feet. The total time for the dive so far was $$\displaystyle{19}{\frac{{{1}}}{{{8}}}}$$. minutes. What was the rate of change in the diver's elevation during the ascent?

2020-11-23
The rate of change in the diver's elevation during the ascent is: rate of change in elevation during descent = time spend ascending/change in elevation during ascent
The change in the diver's elevation during the ascent is the diver's final elevation minus the diver's initial position. The diver's final elevation is $$\displaystyle−{28}{\frac{{{9}}}{{{10}}}}={28.9}$$ feet and the diver's initial evelation is $$\displaystyle-{79}{\frac{{{9}}}{{{10}}}}=-{79.9}$$ feet. The change in the diver's elevation is then:
change in elevation=final elevation−initial elevation
=−28.9−(−79.9) Substitute.
=−28.9+79.9 Simplify.
=79.9−28.9 Rewrite as subtraction.
=51 Subtract.
You know the total time it took the diver to descend, stay at the elevation, and then ascend. The time it took the diver to ascend is then the total time minus the time to descend minus the time spent at the elevation.
You weren't given the time it took the diver to descend but you were given how far the diver descended and at what rate. The diver descended to an elevation of $$\displaystyle-{79}{\frac{{{9}}}{{{10}}}}=-{79.9}$$ feet at a rate of −18.8 feet per minute.
Since distance = (rate)(time), dividing both sides by the rate then gives time=distance/rate.
The time it took the diver to descend is then:
$$\displaystyle{t}{i}{m}{e}={\frac{{−{79.9}}}{{−{18.8}}}}$$ Substitute.
=4.25 Divide.
The time it took the diver to ascend is then the total time of $$\displaystyle{19}{\frac{{{1}}}{{{8}}}}={19.125}$$ minutes minus the time to descend of 4.25 minutes minus the 12.75 minutes the diver spent at an elevation of $$\displaystyle−{79}{\frac{{{9}}}{{{10}}}}$$ ft:
time to ascend =19.125−4.25−12.75
=14.875−12.75 Subtract 19.125 and 4.25.
=2.125 Subtract.
You now have everything you need to find the rate of change in elevation:
rate of change in elevation during descent= time spend ascending/change in elevation during ascent
$$\displaystyle{\frac{{{51}}}{{{2.125}}}}={24}$$
The rate of change in the diver's elevation during the ascent was then 24 feet per minute.

### Relevant Questions

What is the optimal time for a scuba diver to be on the bottom of the ocean? That depends on the depth of the dive. The U.S. Navy has done a lot of research on this topic. The Navy defines the "optimal time" to be the time at each depth for the best balance between length of work period and decompression time after surfacing. Let $$\displaystyle{x}=$$ depth of dive in meters, and let $$\displaystyle{y}=$$ optimal time in hours. A random sample of divers gave the following data.
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critical $$\displaystyle{t}=$$
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