The manufacturer of a certain type of new cell phone battery claims that the average life span of the batteries is 500 charges;
that is, the battery can be charged at least 500 times before failing. To investigate the claim, a consumer group will select a random sample of cell phones with the new battery and use the phones through 500 charges of the battery.a. If the claim is true, what is P( X ≤ 36.7) ?b. Based on the answer to part (a), if the claim is true, is a sample mean lifetime of 36.7 hours unusually short?c. If the sample mean lifetime of the 100 batteries were 36.7 hours, would you find the manufacturer's claim to be plausible? Explain.d. If the claim is true, what is P( X ≤ 39.8)?e. Based on the answer to part (d), if the claim is true, is a sample mean lifetime of 39.8 hours unusually short?f. If the sample mean lifetime of the 100 batteries were 39.8 hours, would you find the manufacturer's claim to be plausible? Explain.
1 answer:
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Answer:
Impossible. t=30 minutes.
Step-by-step explanation:
We have the function:
Where v(t) represents the amount of gallons remaining after t minutes.
We want to find at what time t is the <em>instanteous </em>rate of change from the tank 1000 gallons per minute.
In order to determine the instantaneous rate of change, let's find the derivative of our function. So, take the derivative of both sides with respect to our time t:
On the right, let's move the coefficient outside:
To differentiate, let's use the chain rule, which is:
Our u(x) is x² and v(x) is (1-t/20). So, u'(x) is 2x and v'(x) is -1/20. Therefore:
Simplify:
Simplify:
Distribute:
So, the instantaneous rate of change after time t is given by the above function.
To find when the instantaneous rate of change of the water is 1000 gallons per minute, substitute 100 for v'(t) and solve for t. So:
Solve for t. Add 2000 to both sides:
Divide both sides by 100:
So, after 30 minutes, the instantaneous rate of change will be 1000 gallons per minute.
However, if we go back to our original function, our domain t is only defined between 0 and 20 minutes.
So, it is impossible for our instantaneous rate of change to ever reach 1000 gallons per minute.