Exam 5: Exponential and Logarithmic Functions

The strain (percent of compression) on the lumbar vertebral disks in an adult human as a function of the load x (in kilograms) is given by . ​ Give the answer to four decimal places. ​ What is the rate of change of the strain with respect to the load when the load is 100 kg? ​ __________ ​ What is the rate of change of the strain with respect to the load when the load is 350 kg? ​ __________

Solve the equation for . ​

Use the laws of logarithms to simplify the expression. ​ ​

The percent of a certain brand of computer chips that will fail after t years of use is estimated to be ​ . ​ What percent of this brand of computer chips are expected to be usable after 6 years? ​

On the Richter scale, the magnitude R of an earthquake is given by the formula where I is the intensity of the earthquake being measured and I₀ is the standard reference intensity. ​ Express the intensity I of an earthquake of magnitude in terms of the standard intensity I₀. ​ Express the intensity I of an earthquake of magnitude in terms of the standard intensity I₀. ​ How many times greater is the intensity of an earthquake of magnitude 8 than one of magnitude 6? ​ In modern times the greatest loss of life attributable to an earthquake occurred in eastern China in 1976. Known as the Tangshan earthquake, it registered 8.2 on the Richter scale. How does the intensity of this earthquake compare with the intensity of an earthquake of magnitude ? Round your answer to the nearest integer. ​

Find the derivative of the function. ​ ​

The U.S. population is approximated by the function , where P(t) is measured in millions of people and t is measured in 30-year intervals, with t = 0 corresponding to 1930. What is the expected population of the United States in 2020 (t = 3)? Round your answer to one decimal place. ​ __________ million

Find the derivative of the function. ​

An investor purchased a piece of waterfront property. Because of the development of a marina in the vicinity, the market value of the property is expected to increase according to the rule where V(t) is measured in dollars and t is the time in years from the present. If the inflation rate is expected to be 10% compounded continuously for the next 8 years, what is P(t) expected to be in 4 years? Round your answer to the nearest dollar. ​

Use the laws of logarithms to simplify the expression. ​

The concentration of a drug in grams/cubic centimeter (g/cm3) t minutes after it has been injected into the bloodstream is given by where w, p, and k are positive constants, with p > w. ​ At what time is the concentration of the drug the greatest? What will be the concentration of the drug in the long run? ​

Sketch the graphs of the given functions on the same axes. ​ and ​

A certain piece of machinery was purchased 5 years ago by Garland Mills for $400,000. Its present resale value is $256,000. Assuming that the machine's resale value decreases exponentially, what will it be 2 years from now? Round your answer to the nearest dollar. ​ $__________

The Estradas are planning to buy a house 2 year(s) from now. Housing experts in their area have estimated that the cost of a home will increase at a rate of 9% / year during that 2-year period. If this economic prediction holds true, how much can they expect to pay for a house that currently costs $90,000? Round your answer to two decimal places. ​

Use the curve-sketching guideline, to select the graph of the function. ​ ​

If the temperature is constant, then the atmospheric pressure P (in pounds/square inch) varies with the altitude above sea level h in accordance with the law where p​₀ is the atmospheric pressure at sea level and k is a constant. ​ If the atmospheric pressure is 17 lb/in.² at sea level and 11.5 lb/in.² at 5,000 ft, how fast is the atmospheric pressure changing with respect to altitude at an altitude of 15,000 ft? Give the answer to four decimal places.

The growth rate of the bacterium, a common bacterium found in the human intestine, is proportional to its size. Under ideal laboratory conditions, when this bacterium is grown in a nutrient broth medium, the number of cells in a culture doubles approximately every 20 min. If the initial cell population is 10, determine the function Q(t) that expresses the exponential growth of the number of cells of this bacterium as a function of time t (in minutes). How long will it take for a colony of 10 cells to increase to a population of 1 million? If the initial cell population were 100, how would this alter the model? ​

​Express the equation in logarithmic form. ​

Find the derivative of the function. ​ ​

Find the derivative of the function. ​ ​