Logarithmic Function

Data Modeling

Practice Set 1

Problems 1- 13, solve each question.

1. Find the velocity of a spacecraft whose booster rocket has a mass ratio of R = 20, an exhaust velocity of 2.7 km/s and a firing time of t = 30 s. Use v(t) = −0.0098t + c ln R. Assuming a stable orbit requires minimum velocity of 7.7 km/s, can the spacecraft achieve a stable orbit 300 km above the Earth?

2. Use v = −0.0098t + c ln R. A rocket has a mass ratio of 25 and an exhaust velocity of 2.5 km/s.

Determine the minimum firing time for a stable orbit 300 km above Earth.

3. Find the magnitude of the Mexico City earthquake in 1985 with a seismograph reading of 125,892 millimeters 100 kilometers from the center. R(x) = log (x0/x)

4. How many times more severe was the 2021 earthquake in Fukushima, Japan (R1 = 9.0) than the 2022 earthquake in Fukushima, Japan (R1 = 7.4)?

5. The pH scale for acidity is defined by pH= − log[H+] where [H+] is the concentration of hydrogen ions measures in moles per liter (M). Calculate the concentration of hydrogen ions in moles per liter for a solution that has a pH of 8.75.

6. The pH of a carbonated soda is 3.9 and the pH of ammonia is 11.9.

A. Find the hydrogen-ion concentrations for each liquid.

B. How many times greater is the hydrogen-ion concentration of the soda than that of the ammonia?

C. By how many orders of magnitude do the concentrations differ?

7. New components reduce the sound intensity of a certain model of leaf blower from 10−5 W/m2 to 6.73 × 10−7 W/m2. By how many decibels do these new components reduce the leaf blower’s loudness?

8. A company manufactures noise reducing earbuds for sleeping. Quiet-Nite claims to block sounds as loud as 22 dB. Engineers want to make a model that will reduce the sound intensity to 25% of the Quiet-Nite earbuds. By how many decibels would the loudness be reduced? (Use I0 = 10−12 W/m2 .) Note: See example #4 formula.

9. The Richter scale is used to measure earthquakes. The magnitude of an earthquake is modeled by the equation R = 0.67 log 0.37E + 1.46 , where E is the energy in kilowatt-hours, released by the earthquake.

A. How many kilowatt-hours of energy is released in an earthquake that measures 7.5 on the Richter scale?

B. Find the magnitude of an earthquake that releases 1.5 × 1010 kilowatt-hours of energy.

10. A pizza baked in an over at 475℉ is removed from the oven into a room that is a constant 75℉. After 10 minutes the pizza temperature is at 350℉.

A. Find the time it takes for the pizza to reach 175℉.

B. If you want to eat the pizza when the temperature is 145℉, how long will you wait?

11. In 1989, a San Francisco construction crew unearthed skeletal remains. It was shown that the skeletons had 88% of the expected amount of carbon-14 that is found in a living person. By using the exponential decay model for carbon-14, predict how old the skeletons were in 1989.

Exponential decay model: A(t) = A0e−0.000121t

12. The Dead Sea Scrolls were found in 1947 by an Arab herdsman. Archaeologists determined that the sample of the scrolls contained 76% of their original amount of carbon-14. If the half-life of carbon-14 is 5730 years, estimate the age of the Dead Sea Scrolls.

13. Mrs. Richards’ students took a PreCalculus test and then tested every month with a similar exam.

The average scores for class can be modeled by f(t) = 80 − 17 log t + 1 for 0 ≤ t ≤ 12 , where t is the time in months.

A. Find the average score of students on the original test.

B. Find the average score after 6 months.