4.8 Fitting exponential models to data  (Page 11/12)

The population of a city is modeled by the equation $P(t)=256,114e^{0.25t}$ where $t$ is measured in years. If the city continues to grow at this rate, how many years will it take for the population to reach one million?

Find the inverse function $f^{-1}$ for the exponential function $f\left(x\right)=2\cdot e^{x+1}-5.$

Find the inverse function $f^{-1}$ for the logarithmic function $f\left(x\right)=0.25\cdot {\log }_2\left(x^3+1\right).$

To the nearest minute, what is the half-life of the drug?

Write an exponential model representing the amount of the drug remaining in the patient’s system after $t$ hours. Then use the formula to find the amount of the drug that would remain in the patient’s system after $24$ hours. Round to the nearest hundredth of a gram.

Use Newton’s Law of Cooling to write a formula that models this situation.

How many minutes will it take the soup to cool to $\text{85°F?}$

How many people started the rumor?

To the nearest tenth, how many days will it be before the rumor spreads to half the carrying capacity?

What is the carrying capacity?

Find a formula for an exponential equation that goes through the points $\left(-2,100\right)$ and $\left(0,4\right).$ Then express the formula as an equivalent equation with base e.

What is the carrying capacity for a population modeled by the logistic equation $P(t)=\frac{250,000}{1+499e^{-0.45t}}?$ What is the initial population for the model?

The population of a culture of bacteria is modeled by the logistic equation $P(t)=\frac{14,250}{1+29e^{-0.62t}},$ where $t$ is in days. To the nearest tenth, how many days will it take the culture to reach $75\%$ of its carrying capacity?