MS 101 Alg & Trig test two Name:

15 June 2012

1. _______________ Evaluate log₁₅(50625).
2. ______________________________ Expand ln(xyz²) if x, y and z are real and positive.
3. _______________ Solve 4(7^x) = 9604.
4. _______________ Solve $\ln \sqrt{x+2}=2$. Give the approximate form (the numeric result, keep four decimals). This is "toughie", think about using a fractional exponent...
5. Mathematical models!
   
   The chart above depicts three possible models for a sprinter running the 100 meter dash. As listed in the chart legend from top to bottom, the models are exponential, logarithmic, and logistic. The command to generate this plot for yourself is:
   plot e^(0.025*x),2.6*ln(x),-12+24/(1+e^(-0.15*x)) from x=0 to 100. .
   The x-axis is the distance in meters from the starting line, the y-axis is the speed of the sprinter in meters per second.
   Based on presentations in class, which of the three mathematical equations best models the distance verus speed for a sprinter?
   
   
6. _______ Using the best mathematical model for a sprinter that you chose above, calculate the speed of the sprinter at a distance of ten meters.
7. _______ Using the best mathematical model for a sprinter, calculate the distance at which the sprinter has a speed of 10 m/s.
8. The soil on Bikini atoll in the Marshall Islands is contaminated by the radioactive element Cesium-137. Like all radioactive elements, Cesium-137 will eventually decay into non-radioactive elements. The rate of decay is an exponential decay given by the formula N = P e^{− 0.023 t} where P is the starting amount of Cesium in the soil in kilograms, t is the number of years, and N is the remaining amount of radioactive Cesium after that number of years.
   __________ If there are 100 kilograms of Cesium-137 in the soil now, how many kilograms will be in the soil in 30 years?
9. ____________ How many years until only one kilogram of the original 100 kilograms of Cesium-137 is left in the soil?
10. According to the Asphalt Institute in Lexington Kentucky, road asphalt quality decays logarithmically. A 1999 report records the decay in a measurement called the Present Condition Rating. When new, road asphalt has a condition rating of 100. Resurfacing is expected to occur when the PCR is between 70 and 80. Major road rehabilitation is expected to occur when the asphalt has decayed to a PCR of 60
    

    Age of asphalt in years	Present Condition Rating
    0	100
    10	85
    30	65
    50	57

    
    Use LibreOffice.org to plot the data on an xy scattergraph. Add a logarithmic trend line. Write the logarithmic equation that best fits the data (round numbers to two decimal places):
    
    
    
11. __________ Use the equation obtained from LibreOffice.org to determine the PCR for a 15 year old road.
12. __________ Use the equation obtained from LibreOffice.org to determine the age when a road will reach a PCR of 70 and require resurfacing.
13. __________ The data in the study is based on measurements made on roads in the United States. Would the same decay curve apply to the asphalt roads in Micronesia?
14. Support your answer to the question above: why or why not?