2.7 Mathematical Models

The amount, in milligrams, of a medicinal drug in the body `t` hours after it was injected is given by `D(t)=23(0.85)^t, t >= 0.` Before this injection, the amount of the drug in the body was zero. 

(a) Write down

(i) the initial dose of the drug. 

(ii) the percentage of the drug that leaves the body each hour. 

(b) Calculate the amount of the drug remaining in the body 10 hours after the injection.

(a) 

(i) 23 mg

(ii) 0.15

15 (%)

(b) `23(0.85)^10`

4.53 mg (4.52811...)

Professor Wei observed that students have difficulty remembering the information presented in his lectures.

He modelled the percentage of information retained, `R`, by the function `R(t)=100e^{-pt}, t >= 0,` where `t` is the number of days after the lecture.

He found that 1 day after a lecture, students had forgotten 50% of the information presented.

(a) Find the value of `p`.

(b) Use this model to find the percentage of information retained by his students 36 hours after Professor Wei’s lecture. 

Based on his model, Professor Wei believes that his students will always retain some information from his lecture.

(c) State a mathematical reason why Professor Wei might believe this. 

(d) Write down one possible limitation of the domain of the model. 

(a) 

0.693 (0.693147..., ln 2)

(b) `R(1.5)=100e^{-0.693147... times 1.5}`

35.4(%) (35.3553...)

(c) `R(t) > 0`

(d) Award for one reasonable limitation of the domain: 

• small values of `t` produce unrealistic results
• `R`(0) = 100%
• large values of `t` are not possible
• people do not live forever
• model is not valid at small or large values of `t`

The reason should focus on the domain `t` > 0. Do not accept answers such as:

• recollection varies for different people
• memories are discrete not continuous
• the nature of the information will change how easily it is recalled
• emotional/physical stress can affect recollection/concentration

Natasha carries out an experiment on the growth of mould. She believes that the growth can be modelled by an exponential function

`P(t)=Ae^{kt}`

where `P` is the area covered by mould in mm², `t` is the time in days since the start of the experiment and `A` and `k` are constants.

The area covered by mould is 112 mm² at the start of the experiment and 360 mm² after 5 days.

(a) Write down the value of `A`.

(b) Find the value of `k`.

(a) `A` = 112

(b) `112e^{5k}=360`

`(k=) 1/5 ln(360/112)`

`(k=) 0.234 \ (0.233521...)`

The pH of a solution measures its acidity and can be determined using the formula pH = `-log_10 C`, where `C` is the concentration of hydronium ions in the solution, measured in moles per litre. A lower pH indicates a more acidic solution.

The concentration of hydronium ions in a particular type of coffee is `1.3 × 10^-5` moles per litre.

(a) Calculate the pH of the coffee. 

A different, unknown, liquid has 10 times the concentration of hydronium ions of the coffee in part (a).

(b) Determine whether the unknown liquid is more or less acidic than the coffee. Justify your answer mathematically. 

(a) (pH =) `-log_10 (1.3 times 10^-5)`

4.89 (4.88605...)

(b) calculating pH

(pH =) `-log_10 (10 times 1.3 times 10^-5)`

3.89 (3.88605...)

(3.89 < 4.89, therefore) the unknown liquid is more acidic (than coffee).

The strength of earthquakes is measured on the Richter magnitude scale, with values typically between 0 and 8 where 8 is the most severe. The Gutenberg–Richter equation gives the average number of earthquakes per year, `N`, which have a magnitude of at least `M`. For a particular region the equation is 

`log_10 N = a - M`, for some `a in RR.`

This region has an average of 100 earthquakes per year with a magnitude of at least 3.

(a) Find the value of `a`.

The equation for this region can also be written as `N=b/10^M`.

(b) Find the value of `b`.

(c) Given 0 < `M` < 8, find the range for `N`. 

The expected length of time, in years, between earthquakes with a magnitude of at least `M` is `1/N`.

Within this region the most severe earthquake recorded had a magnitude of 7.2.

(d) Find the expected length of time between this earthquake and the next earthquake of at least this magnitude. Give your answer to the nearest year. 

(a) `log_10 100=a-3`

`a=5`

(b) `100=b/10^3`

`b=100000 (=10^5)`

(c) 0.001< `N` <100000(10^-3 < `N` < 10⁵) 

(d) `N=10^5/10^7.2 \ (=0.0063095 ...)`

length of time = `1/{0.0063095 ...}=10^2.2`

= 158 years

A cup of hot water is placed in a room and is left to cool for half an hour. Its temperature, measured in °C, is recorded every 5 minutes. The results are shown in the table.

Akira uses the power function `T(t)=at^b+25` to model the temperature, `T`, of the water `t` minutes after it was placed in the room. 

(a) State what the value of 25 represents in this context. 

(b) Use your graphic display calculator to find the value of `a` and of `b`. 

Soo Min models the temperature, `T`, of the water `t` minutes after it was placed in the room as `T(t)=k c^t+25.`

(c) Find the value of `k` and of `c`. 

(d) State a reason why Soo Min’s model of the temperature is a better fit for the data than Akira’s model. 

(a) room temperature / the temperature below which the hot water will not cool 

(b) evidence of subtracting 25 from the temperature data 

`a` = 244 (243.920 ... ) and `b` = −1.03 (−1.02965)

(c) `k` = 61.1 (61.0848 ... ) and `c` = 0.923 (0.923029 ... ) 

(d) consider value at `t` = 0 / “water cannot reach a temperature more than 100 degrees” 

A machine was purchased for $10000. Its value `V` after `t` years is given by `V=10000e^{-0.3t}.` The machine must be replaced at the end of the year in which its value drops below $1500. Determine in how many years the machine will need to be replaced. 

`10000e^{-0.3t}=1500`

The population `p` of bacteria at time `t` is given by `p=100e^{0.05t}`. 

Calculate

(a) the value of `p` when `t` = 0; 

(b) the rate of increase of the population when `t` = 10. 

(a) `p=100e^0=100`

(b) Rate of increase is `{dp}/{dt}`

`{dp}/{dt}=0.05 times 100 e^{0.05t} \ (=5e^{0.05t})`

When `t=10`

`=5e^{0.05(10)}=5e^0.5 \ (=8.24=5 sqrte )`

Elvis Presley is an extremely popular singer. Although he passed away in 1977, many of his fans continue to pay tribute by dressing like Elvis and singing his songs.
The number of Elvis impersonators, `N(t)`, can be modelled by the function

`N(t)=170 times 1.31^t`

where `t`, is the number of years since 1977.

(a) Write down the number of Elvis impersonators in 1977. 

(b) Calculate the time taken for the number of Elvis impersonators to reach 130000. 

(c) Calculate the number of Elvis impersonators when `t` = 70. 

The world population in 2047 is projected to be 9 500 000 000 people.

(d) Use this information to explain why the model for the number of Elvis impersonators is unrealistic. 

(a) 170

(b) 130000 = 170 × 1.31^t

(`t` =) 24.6 (24.5882 ... (years))

(c) 170 × 1.31⁷⁰ 

2.75 × 10¹⁰ (2.75067 ... × 10¹⁰, 27 500 000 000, 27 506 771 343)

(d) The number of Elvis impersonators in 2047 is greater than the world population. 

The car’s value in dollars, `V`, is modelled by the function

`V(t)=12870-k(1.1)^t, t >=0`

where `t` is the number of years since the car was purchased and `k` is a constant.

(a) Write down, and simplify, an expression for the car’s value when Gabriella purchased it. 

After two years, the car’s value is $9143.20.

(b) Find the value of `k`.

This model is defined for `0 <= t <= n`. At `n` years the car’s value will be zero dollars.

(b) Find the value of `n`.

(a) 12870 − `k`(1.1)⁰ = 12870 − `k`

(b) 9143.20 = 12870 − `k`(1.1)²

(`k` =) 3080

(c) 12870 − 3080(1.1)^`n` = 0

(`n` =) 15.0 (15.0033 ... )