Exam 5: Exponential and Logarithmic Functions

Strontium 90, a radioactive isotope of strontium, is present in the fallout resulting from nuclear explosions. It is especially hazardous to animal life, including humans, because, upon ingestion of contaminated food, it is absorbed into the bone structure. Its half-life is 27 years. If the amount of strontium 90 in a certain area is found to be eight times the "safe" level, find how much time must elapse before an "acceptable level" is reached. ​ __________ years

Find the effective rate corresponding to the given nominal rate. Round your answers to the nearest thousandth. ​ 5% / year compounded monthly ​ __________ % ​ 5% / year compounded daily ​ __________ %

Phosphorus 32 has a half-life of 14.2 days. If 600 g of this substance are present initially, find the amount present after t days. What amount will be left after 3.55 days? How fast is the phosphorus 32 decaying when t = 3.55? ​

Find the interest rate needed for an investment of $4,000 to grow to an amount of $8,500 in 3 year(s) if interest is compounded monthly. Round your answer to two decimal places. ​

Use the definition of a logarithm to prove .

Simplify the expression. ​

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

Write the expression as the logarithm of a single quantity. ​ ​

Use logarithmic differentiation to find the derivative of the function. ​ ​

Evaluate the expression. ​ ​

Use the facts that to find the value of the logarithm. ​ ​

Find an equation of the tangent line to the graph of at its inflection point. ​

The price of a certain commodity in dollars per unit at time t (measured in weeks) is given by ​ . ​ What is the price of the commodity at t = 0? ​ $__________ / unit ​ How fast is the price of the commodity decreasing at t = 0? ​ $__________ / wk ​ Find the equilibrium price of the commodity.

The height (in feet) of a certain kind of tree is approximated by where t is the age of the tree in years. Estimate the age of an 75-ft tree. Please round the answer to two decimal places. ​

The unit selling price p (in dollars) and the quantity demanded (in pairs) of a certain brand of women's gloves is given by the demand equation . ​ What is the marginal revenue to the nearest cent per day when x = 100? ​

Bernie invested a sum of money 7 year(s) ago in a savings account, which has since paid interest at the rate of 8% / year compounded quarterly. His investment is now worth $22,927.66. How much did he originally invest? Round your answer to the nearest cent. ​

Use the laws of logarithms to simplify the expression. ​

On the basis of data collected during an experiment, a biologist found that the growth of the fruit fly (Drosophila) with a limited food supply could be approximated by the exponential model where t denotes the number of days since the beginning of the experiment. ​ How fast was the population changing on the 20th day?

The radioactive element polonium decays according to the law where Q​₀ is the initial amount and the time t is measured in days. If the amount of polonium left after 560 days is 40 mg, what was the initial amount present? ​

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 12.5 lb/in.² at 6,000 ft, find the atmospheric pressure at an altitude of 18,000 ft. Please round the answer to one decimal place. ​ __________ lb/in.² ​ How fast is the atmospheric pressure changing with respect to altitude at an altitude of 18,000 ft? Please round the answer to four decimal places. ​ __________ lb/in.²/ft