Exponential and Logarithmic Models (Day 2) 3.5

And the case of the murdered professor Calvin (Cal) Q. Les And the case of the murdered professor

Suspects LaBron James Albert Einstein Seen arguing with Mr. Cutler at 1:30pm Was boarding a plane heading for Cleveland, OH at Sky Harbor at 2:45pm Seen leaving the math building at 3:05pm Was playing at the Discovery Science Center between 10am and 2pm

Suspects Brad Pitt Bill Gates Seen hanging outside Mr. Cutler’s house at 6:15pm Was at an autograph signing from 1pm-5pm Was part of a CHS Engineering ThinkTank group from noon to 9pm Whereabouts unknown from 7pm to 8:15pm

According to Newton’s Law of Cooling, the time that elapses since death can be calculated using the model 𝑡=−10 ln 𝑇− 𝑅 𝑡 𝑁𝐵 𝑡 − 𝑅 𝑡 , where… t is the time that has elapsed in hours T is the Temperature of the body at a given time Rt is the Room temperature NBt is the Normal Body temperature

Just the Facts, Ma’am… Body found at 8:30pm At 9pm, temperature of the body was 85.7℉ At 11pm, temperature of the body was 82.8℉ Room temperature was a constant 70.0℉ all weekend Assume Mr. Cutler was healthy and had a normal body temperature of 98.6℉ at the time of death According to Newton’s Law of Cooling, the time that elapses since death can be calculated using the model 𝑡=−10 ln 𝑇− 𝑅 𝑡 𝑁𝐵 𝑡 − 𝑅 𝑡

Whodunnit?

Objectives Use exponential and logarithmic functions to model and solve real-life problems.

Example 2 – Population Growth In a research experiment, a population of fruit flies increases according to the exponential model 𝑦=33.33 𝑒 𝑘𝑡 . After 2 days there are 100 flies. Find the value of k then determine how many flies there will be after 5 days.

Example 6 – Magnitudes of Earthquakes On the Richter scale, the magnitude R of an earthquake of intensity I is given by where I0 = 1 is the minimum intensity used for comparison. Find the intensity of each earthquake. (Intensity is a measure of the wave energy of an earthquake.) a. Alaska in 2012: R = 4.0 b. Christchurch, New Zealand, in 2011: R = 6.3

Example 6(b) – Solution cont’d Note that an increase of 2.3 units on the Richter scale (from 4.0 to 6.3) represents an increase in intensity by a factor of 2,000,000/10,000 = 200. In other words, the intensity of the earthquake in Christchurch was about 200 times as great as that of the earthquake in Alaska.

3.5 Example – Worked Solutions

Example 2 – Solution In a research experiment, a population of fruit flies increases according to the exponential model 𝑦=33.33 𝑒 𝑘𝑡 . After 2 days there are 100 flies. Find the value of k then determine how many flies there will be after 5 days. 100=33.33 𝑒 𝑘(2) plug in known values 100 33.33 = 𝑒 2𝑘 divide by 33.33 ln 100 33.33 =2𝑘 “Circular Rule of 3” 0.5494≈𝑘 solve for k

Example 2 – Solution After 5 years 𝑦=33.33 𝑒 𝑘𝑡 yields… 𝑦=33.33 𝑒 0.5494(5) 𝑦≈520 flies

Example 6 – Magnitudes of Earthquakes On the Richter scale, the magnitude R of an earthquake of intensity I is given by where I0 = 1 is the minimum intensity used for comparison. Find the intensity of each earthquake. (Intensity is a measure of the wave energy of an earthquake.) a. Alaska in 2012: R = 4.0 b. Christchurch, New Zealand, in 2011: R = 6.3

Example 6(a) – Solution Because I0 = 1 and R = 4.0, you have 4.0 = 104.0 = 10log I 104.0 = I 10,000 = I. Substitute 1 for I0 and 4.0 for R. Exponentiate each side. Inverse Property Simplify.

Example 6(b) – Solution cont’d For R = 6.3, you have 6.3 = 106.3 = 10log I 106.3 = I 2,000,000 ≈ I. Substitute 1 for I0 and 6.3 for R. Exponentiate each side. Inverse Property Use a calculator.

Example 6(b) – Solution cont’d Note that an increase of 2.3 units on the Richter scale (from 4.0 to 6.3) represents an increase in intensity by a factor of 2,000,000/10,000 = 200. In other words, the intensity of the earthquake in Christchurch was about 200 times as great as that of the earthquake in Alaska.