IB Mathematics: Applications and Interpretation Quiz: Exponential And Logarithmic Models

A cup of coffee with an initial temperature of 95°C is left to cool in a room with a constant temperature of 20°C. The temperature, $T$, of the coffee after $t$ minutes is modelled by $T(t) = 20 + 75e^{-kt}$. After 5 minutes, the temperature is 60°C. Find the time it takes for the coffee to cool to 30°C.

The temperature $T$ in degrees Celsius inside a building $t$ hours after a power failure is modelled by $T(t) = 5 + 17e^{-0.08t}$. How long does it take for the temperature to drop by 10°C from its initial temperature at $t=0$?

A scientist models the number of cells in a culture, $N$, after $t$ days with the function $N(t) = 100e^{0.23t}$. Another scientist proposes a model of the form $N(t) = 100(b)^t$. What is the value of $b$ that makes the two models equivalent?

The population of a town, $P$, is modelled by the function $P(t) = 5000 \times (1.02)^{2t}$, where $t$ is the number of years since 2020. Which of the following statements correctly describes the annual percentage growth rate of the population?

The pH of vinegar is 2.2 and the pH of a tomato is 4.7. The pH is defined as $pH = -\log_{10}[H^+]$, where $[H^+]$ is the hydrogen ion concentration. How many times greater is the hydrogen ion concentration in vinegar than in the tomato? Round your answer to the nearest integer.

The value of a car, $V$ in dollars, depreciates according to the model $V(t) = 25000(0.85)^t$, where $t$ is the number of years since it was purchased. What is the value of $\log_{10}(V)$ after 5 years, to two decimal places?

A company finds that its profit $P$ in thousands of dollars is modelled by $P(t) = 50 - 45e^{-0.1t}$, where $t$ is the number of months since its launch. What is the company's profit in the long run, according to this model?

The population of a species of frog is decreasing exponentially. In 2010, the population was 1200. In 2018, the population was 750. Assuming the population follows a model of the form $P(t) = P_0 e^{-kt}$, where $t$ is the number of years since 2000, what was the estimated population in 2000?

The concentration of a drug in a patient's bloodstream, $C$, in mg/L, $t$ hours after administration is modelled by $C(t) = 15e^{-0.2t}$. The drug is effective as long as the concentration is above 2 mg/L. For how many hours is the drug effective? Give your answer to the nearest hour.

The population of a town, $P$, is modelled by the function $P(t) = 5000 \times (1.02)^{2t}$, where $t$ is the number of years since 2020. Which of the following statements correctly describes the annual percentage growth rate of the population?

Anja invests $10,000 in Fund A which earns 4% annual interest compounded continuously. Boris invests $8,000 in Fund B which earns 6% annual interest compounded continuously. After how many years will the value of Boris's investment first equal the value of Anja's investment?

A biological culture contains 3000 bacteria. The population is observed to triple every 4 hours. The population $P$ after $t$ hours is modelled by $P(t) = 3000 \times k^t$. What is the value of $k$, correct to three significant figures?

The magnitude of an earthquake, $M$, on the Richter scale is given by $M = \frac{2}{3}\log_{10}(\frac{E}{E_0})$, where $E$ is the energy released and $E_0$ is a constant. An earthquake with magnitude 6.0 is followed by an aftershock with magnitude 4.0. The energy released by the first earthquake is how many times the energy released by the aftershock?

A scientist models the number of cells in a culture, $N$, after $t$ days with the function $N(t) = 100e^{0.23t}$. Another scientist proposes a model of the form $N(t) = 100(b)^t$. What is the value of $b$ that makes the two models equivalent?

The temperature $T$ in degrees Celsius inside a building $t$ hours after a power failure is modelled by $T(t) = 5 + 17e^{-0.08t}$. How long does it take for the temperature to drop by 10°C from its initial temperature at $t=0$?

A sample of a radioactive isotope initially has a mass of 200 mg. After 10 days, its mass is 150 mg. The decay is modelled by the function $M(t) = 200e^{-kt}$, where $t$ is the time in days. What is the half-life of this isotope, to the nearest tenth of a day?

An ecologist is studying the relationship between the body mass ($M$, in kg) and the brain mass ($B$, in g) of different mammal species. When the ecologist plots $\log_{10}(B)$ against $\log_{10}(M)$, the data points lie close to a straight line with the equation $y = 0.75x + 0.9$. Which of the following functions correctly models the relationship between $B$ and $M$?

The number of views, $V$, of a new online video is modelled by $V(t) = 150 \times (1.3)^t$, where $t$ is the number of hours after it was uploaded. To the nearest minute, how long does it take for the number of views to double?

Anja invests $10,000 in Fund A which earns 4% annual interest compounded continuously. Boris invests $8,000 in Fund B which earns 6% annual interest compounded continuously. After how many years will the value of Boris's investment first equal the value of Anja's investment?

The loudness of a sound, $L$, in decibels (dB), is given by $L = 10 \log_{10}(\frac{I}{I_0})$, where $I$ is the intensity of the sound and $I_0 = 10^{-12} \text{ W/m}^2$. A rock concert has a loudness of 115 dB. Find the intensity, $I$, of the sound at the concert.