IB Mathematics Applications & Interpretation Question Type 3: Determining Best Fit Straight Lines for Linearized Models Exercises

From a best-fit line $\ln \left(\frac{y}{4}\right)=0.22x$, determine the value of $r$ in the model $y=4e^{rx}$, and hence predict $y$ when $x=5$.

A linearized plot of $\ln (y)$ versus $x$ yields slope $r=0.2$. Determine the doubling time of $y$ in the model $y=4e^{rx}$.

From a plot of $\ln \left(\frac{y}{4}\right)$ versus $x$, the best-fit line is $\ln \left(\frac{y}{4}\right)=0.35x-0.02$. Determine $r$ in the model $y=4e^{rx}$ and interpret the small intercept.

In the model $y=4e^{rx}$, you observe $y=20$ when $x=10$. Find $r$ and then determine $y$ at $x=2$.

Given that in the model $y=4e^{rx}$ one measurement is $y=10$ at $x=5$, solve for $r$.

A radioactive substance follows $y=4e^{-rx}$. A plot of $\ln (y/4)$ versus $x$ has slope $-0.07$. Find the decay constant $r$ and the half-life of the substance.

Given data points $(x,y)=(0,4),(2,6.0),(4,9.0)$ and assuming $y=4e^{rx}$, linearize the relationship by plotting $\ln (y/4)$ against $x$, then use the two points $(2,\ln (6/4))$ and $(4,\ln (9/4))$ to determine $r$.

A best-fit line for $\ln (y)$ versus $x$ gives the equation $$\ln (y)=0.4x+\mathrm{1.3863.}$$ Assuming the model $y=4e^{rx}$, determine the value of $r$.

Experimental data yields the best-fit line $$\ln (y)=0.27x+1.2$$ for a model $y=4e^{rx}$. Find $r$ and compare the intercept with the theoretical value $\ln (4)$.

A plot of $\ln (y/4)$ versus $x$ yields a slope $r=0.15$. How long must one wait for $y$ to reach 10 in the model $y=4e^{rx}$?

A linearized plot of $\ln (y)$ versus $x$ yields slope $r=-0.1$. Calculate the half-life of $y$ if $y=4e^{rx}$.

A population starts at $y=4$ and doubles every 3 hours. Write a model of the form $y=4e^{rx}$ and find the constant $r$.