Exam 6: Section 2: Differential Equations

The half life of the radium isotope Ra-226 is approximately 1,599 years. If the amount left after 1,000 years is 1.8 g, what is the amount after 2000 years? Round your answer to three decimal places.

The isotope has a half-life of 5,715 years. After 2,000 years, a sample of the isotope is reduced to 1.2 grams. What was the initial size of the sample (in grams)? How much will remain after 20,000 years (i.e., after another 18000 years)? Round your answers to four decimal places.

Find the principal that must be invested at the rate 8%, compounded monthly, so that $1,000,000 will be available for retirement in 50 years. Round your answer to the nearest cent.

The rate of change of N is proportional to N. When and when . What is the value of N when ? Round your answer to three decimal places.

Solve the differential equation.

The isotope has a half-life of 5,715 years. Given an initial amount of 11 grams of the isotope, how many grams will remain after 500 years? After 5,000 years? Round your answers to four decimal places.

Write and solve the differential equation that models the following verbal statement: The rate of change of with respect to is proportional to .

Solve the differential equation.

Find the function passing through the point with the first derivative .

Solve the differential equation.

Find the time (in years) necessary for 1,000 to double if it is invested at a rate 6% compounded continuously. Round your answer to two decimal places.

The initial investment in a savings account in which interest is compounded continuously is $604. If the time required to double the amount is years, what is the amount after 15 years? Round your answer to the nearest cent.

Find the function passing through the point with the first derivative .

The initial investment in a savings account in which interest is compounded continuously is $813. If the time required to double the amount is years, what is the annual rate? Round your answer to two decimal places.

Suppose that the population (in millions) of Paraguay in 2007 was 6.7 and that the expected continuous annual rate of change of the population is 0.024. Find the exponential growth model for the population by letting correspond to 2000. Round your answer to four decimal places.

Suppose that the population (in millions) of a Egypt in 2007 is 80.3 and that expected continuous annual rate of change of the population is 0.017. The exponential growth model for the population by letting corresponds to 2000 is . Use the model to predict the population of the country in 2013. Round your answer to two decimal places.

The isotope has a half-life of 24,100 years. After 10,000 years, a sample of the isotope is reduced 1.6 grams. What was the initial size of the sample (in grams)? How large was the sample after the first 1,000 years? Round your answers to four decimal places.

The half-life of the radium isotope Ra-226 is approximately 1,599 years. If the initial quantity of the isotope is 38 g, what is the amount left after 1,000 years? Round your answer to two decimal places.

The number of bacteria in a culture is increasing according to the law of exponential growth. After 5 hours there are 175 bacteria in the culture and after 10 hours there are 425 bacteria in the culture. Answer the following questions, rounding numerical answers to four decimal places. (i) Find the initial population. (ii) Write an exponential growth model for the bacteria population. Let t represent time in hours. (iii) Use the model to determine the number of bacteria after 20 hours. (iv) After how many hours will the bacteria count be 15,000?

Find the exponential function that passes through the two given points. Round your values of C and k to four decimal places.