111. Cold Water Survival When a person falls through the ice on a lake, the cold water removes body heat 25 times faster than air at the same temperature. To survive, a person has between 30 and 90 seconds to get out of the water. How long could a person remain in air at the same temperature?
81. Aging More Slowly Albert Einstein in his theory of relativity showed that if a person travels at nearly the speed of light, then time slows down significantly. Suppose that there are twins; one remains on Earth and the other leaves in a very fast spaceship having velocity v. If the twin on Earth ages To years, then according to Einstein the twin in the spaceship ages years, where
T(v) = To sqrt(1-(v/c)^2))
In this formula c represents the speed of light, which is 186,000 miles per second.
21. Heart Rate of an Athlete The following table lists the heart rate or pulse of an athlete running a 400-
meter race. The race lasts 50 seconds.
Time (seconds) 0 20 30 50
Heart Rate (bpm) 100 134 150 180
(a)Does model the data in the table exactly? Explain.
Pt = - (1/300)t^2 +(53/30)t + 100
(b) Does P*(60) have significance in this situation? What should be the domain of P?
69. Motor Vehicle Fatalities The number of motor vehicle deaths increased by 13.75 times from 1912 to 2003. There were 39,525 more deaths in 2003 than in 1912. Find the number of motor vehicle deaths in each of the two years. Note that the number of motor vehicles on the road increased from 1 million to 230 million between 1912 and 2003. (Source: Department of Health and Human Services.)
111. To find out how long a person could remain in air at the same temperature, we can use the information given that the cold water removes body heat 25 times faster than air at the same temperature. So if a person has between 30 and 90 seconds to get out of the water, we can divide this range by 25 to find the equivalent range for air.
If we divide 30 seconds by 25, we get 1.2 seconds, and if we divide 90 seconds by 25, we get 3.6 seconds.
Therefore, a person could remain in air at the same temperature for anywhere between 1.2 seconds and 3.6 seconds.
81. According to Albert Einstein's theory of relativity, if a person travels at nearly the speed of light, time slows down significantly. The formula to calculate the time dilation is given as follows:
T(v) = To * sqrt(1 - (v/c)^2)
In the formula, T(v) represents the time experienced by the twin in the spaceship, To represents the time experienced by the twin on Earth, v represents the velocity of the spaceship, and c represents the speed of light.
To find out how much the twin in the spaceship ages, we need to know the value of v. Without that information, we cannot calculate the specific age.
21. To analyze whether the given model represents the data in the table exactly, we need to substitute the values from the table into the given equation and check if it matches.
Plugging in the provided values:
For t = 0, P(t) = - (1/300)(0)^2 + (53/30)(0) + 100 = 100
For t = 20, P(t) = - (1/300)(20)^2 + (53/30)(20) + 100 ≈ 134
For t = 30, P(t) = - (1/300)(30)^2 + (53/30)(30) + 100 ≈ 150
For t = 50, P(t) = - (1/300)(50)^2 + (53/30)(50) + 100 ≈ 180
Comparing these calculated values with the data in the table, we can see that the model does fit the data exactly.
(b) P*(60) refers to the value of P(t) when t is equal to 60 seconds. In this situation, it would represent the heart rate of the athlete 60 seconds into the race. The significance of this value would depend on the context and purpose of the analysis.
The domain of P, in this case, would be the range of values of t that are meaningful in the given situation. Since the race lasts 50 seconds, the domain of P would likely be from 0 to 50 seconds.
69. To find the number of motor vehicle deaths in each of the two years, we can set up a proportion based on the given information.
Let x represent the number of deaths in 1912.
Then, the number of deaths in 2003 would be 13.75x.
According to the given information, the number of deaths in 2003 is 39,525 more than in 1912. So we can set up the equation:
13.75x - x = 39,525
Simplifying the equation:
12.75x = 39,525
Dividing both sides by 12.75:
x = 3,100
Therefore, the number of motor vehicle deaths in 1912 was 3,100, and in 2003 it was 13.75 * 3,100 = 42,625.