Concept explainers
The Marriage Problem There is an infamous problem from mathematics that attempts to quantify the number of potential mates one should date before choosing one’s “true love.” The function
represents the probability of finding the ideal mate after rejecting the first
Determine and interpret
Determine and interpret
What is the domain of
Graph
Judging on the basis of the approach suggested by the model, what is the value of
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Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
- pH of Saliva The pH of saliva is normally in the range of 6.4 to 7.0. However, when a person is ill, the persons saliva becomes more acidic. a When Marco is sick, he tests the pH of his saliva and finds that it is 5.5. What is the hydrogen ion concentration of his saliva? b Will the hydrogen ion concentration in Marcos saliva increase or decrease as he gets better? c After Marco recovers, he tests the pH of his saliva, and it is 6.5. Was the saliva more acidic or less acidic when he was sick?arrow_forwardAn Uncertain Investment Suppose you invested 1300 in the stock market two years ago. During the first year the value of the stock increased by 12%. During the second year, the value of the stock decreased by 12%. How much money is your investment worth at the end of the two-year period? Did you earn money or lose money? Note: The answer to the first question is not 1300arrow_forwardHarvard Step Test The Harvard Step Test was developed4 in 1943 as physical fitness test, and modifications of it remain in use today. The candidate steps up and down on a bench 20 inches high 30 times per minute for 5 minutes. The pulse is counted three separate times for 30 seconds each: at 1 minute, 2 minutes, and 3 minutes after the exercise is completed. If P is the sum of the three pulse counts, then the physical efficiency index E is calculated using E=15,000P. The following table shows how to interpret the results of the test. Efficiency Index Interpretation Below 55 Poor condition 55 to 64 Low average 65 to 79 High average 80 to 89 Good 90 & above Excellent a. Does the physical efficiency index increase or decrease with increasing values of P? Explain in practical terms what this means. b. Express using functional notation the physical efficiency index of someone whose total pulse count is 200, and then calculate that value. c. What is the physical condition of someone whose total pulse cont is 200? d. What pulse counts will result in an excellent rating?arrow_forward
- Radioactive Decay The half-life of a radioactive substance is the time H that it takes for half of the substance to change form through radioactive decay. This number does not depend on the amount with which you start. For example, carbon-14 is known to have a half-life of H=5770 years. Thus, if you begin with 1 gram of carbon-14, then 5770 years later you will have 12 gram of carbon-14. And if you begin with 30 grams of carbon-14, then after 5770 years there will be 15 grams left. In general, radioactive substances decay according to the formula A=A00.5tH Where H is the half-life, t is the elapsed time, A0 is the amount you start with the amount when t=0, and A is the amount left at time t. a. Uranium-228 has a half-life H of 9.3 minutes. Thus, the decay function for this isotope of uranium is A=A00.5t9.3, where t is measured in minutes. Suppose we start with grams of uranium-228. i. How much uranium-228 is left after 2 minutes? ii.How long will you have to wait until there are only 3 grams left? b. Uranium-235 is the isotope of uranium that can be used to make nuclear bombs. It has a half-life of 713 million years. Suppose we start with 5 grams of uranium-235. i. How much uranium-235 is left after 200 million years? ii. How long will you have to wait until there are only 3 grams left?arrow_forwardA Population of Foxes A breeding group of foxes is introduced into a protected area, and the population growth follws a logistic pattern. After t years, the population of foxes is given by N=37.50.25+0.76t foxes. a. How many foxes were intorduced into the protected area? b. Make a graph of N versus t and explain in words how the populatoin of foxes increases with time. c. When will the fox population reach 100 individuals?arrow_forwardFalling Ball Using calculus, it can be shown that if a ball is thrown upward with an initial velocity of 16 ft/s from the top of a building 128 ft high, then its height h above the ground t seconds later will be h=128+16t-16t2 During what time interval will the ball be at least 32 ft above the ground?arrow_forward
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