Problems+from+the+Book

290 || **4** 5 || //5.8 Applications of Logarithms pg. 290-291// 4) The length of time that milk (and many other perishable substances) will stay fresh depends on the storage temperature. Suppose that milk will stay fresh for 146 hours in a refrigerator at 4 degrees Celsius. Milk that is left out in the kitchen at 22 degrees Celsius will keep for only 42 hours. Because bacteria grow exponentially, you can assume that freshness decays exponentially.
 * Page || Problem Number ||
 * 266 || Investigation 1-7 ||
 * 273 || Investigation 1-8 ||
 * 282 || Properties of Logs ||
 * **290**
 * **290**
 * 290 || 6 ||
 * **291** || **7** ||
 * **291** || **8** ||
 * **291** || **9** ||
 * **291** || **10** ||

a. Write an equation that expresses the number of hours, //h//, that milk will keep in terms of the temperature, //T//.

b. Use your equation to predict how long milk will keep at 30 degrees Celsius and at 16 degrees Celsius.

c. If a container of milk soured after 147 hours, what was the temperature at which it was stored?

d. Graph the relationship between hours and temperature, using your equation from 4a and the five data points you have found.

e. What is a realistic domain for this relationship? Why?


 * 7)** The table gives the loudness of spoken words, measured at the source, and the maximum distance at which another person can recognize the speech. Find an equation that expresses the maximum distance as a function of loudness.

a. Plot the data on your calculator and make a rough sketch on your paper.
 * = Loudness (dB) ||= Distance (m) ||
 * = 0.5 ||= 0.1 ||
 * = 3.2 ||= 16.0 ||
 * = 5.3 ||= 20.4 ||
 * = 16.8 ||= 30.5 ||
 * = 35.2 ||= 37.0 ||
 * = 84.2 ||= 44.5 ||
 * = 120.0 ||= 47.6 ||
 * = 170.0 ||= 50.6 ||

b. Experiment to find the relationship between //x// and //y// by plotting different combinations of //x//, //y//, //log x//, and //log y// until you have found the graph that best linearizes the data. Sketch this graph on your paper and label the axes with //x, y, log x,// and //log y//, as appropriate.

c. Find the equation of a line that fits the plot you chose in 7b. Remember that your axes did not represent //x// and //y//, so substitute (log //x//) or (log //y//) into your equation as appropriate.

d. Graph this new equation with the original data. Does it seem to be a good model?

a. Plot the data using an appropriate window. Make a rough sketch of this graph.
 * 8)** A container of juice is left in a room temperature of 74 degrees F. After 8 minutes, the temperature is recorded at regular intervals.
 * = Time (min) ||= 8 ||= 10 ||= 12 ||= 14 ||= 16 ||= 18 ||= 20 ||= 22 ||= 24 ||= 26 ||= 28 ||= 30 ||
 * = Temperature (degrees F) ||= 35 ||= 40 ||= 45 ||= 49 ||= 52 ||= 55 ||= 57 ||= 60 ||= 61 ||= 63 ||= 64 ||= 66 ||

b. Find an exponential model for temperature as a function of time. (//Hint:// This curve is both reflected and translated.)


 * 9)** In clear weather, the distance you can see from a window on a plane depends on your height above Earth, as shown in the table below.

a. Graph various combinations of //x, y, log x,// and //log y// until you find a combination that linearizes the data.

b. Use your results from 9a to find a best-fit equation for data in the form (//height, view//) using the data in this table.
 * ||= Height (m) ||= Viewing Distance (km) ||
 * = 305 ||= 62 ||
 * = 610 ||= 88 ||
 * = 914 ||= 108 ||
 * = 1,524 ||= 139 ||
 * = 3,048 ||= 197 ||
 * = 4,572 ||= 241 ||
 * = 6,096 ||= 278 ||
 * = 7,620 ||= 311 ||
 * = 9,144 ||= 340 ||
 * = 10,668 ||= 368 ||
 * = 12,192 ||= 393 ||
 * 10)** Quinn starts treating her pool for the season with a shock treatment of 4 gal of chlorine. Every12 h, 15% of the chlorine evaporates. The next morning, she adds 1 qt (one-fourth a gal) of chlorine to the pool, and she continues to do so each morning.

a. How much chlorine is there in the pool after one day (after she adds the first daily quart of chlorine)? After two days? Write a recursive formula for this pattern.

b. Use the formula from 10a to make a table of values and sketch a graph of 20 terms. Find an explicit model that fits the data.