Question Type 1: Scaling data appropriately under the context Exercises

Convert the masses $50\text{ g}$, $1500\text{ g}$ and $25000\text{ g}$ to kilograms. Define $M=\dfrac{m}{1000}$ where $m$ is in grams. What are the values of $M$?

A process takes times $t=1800\,$s, $5400\,$s and $9000\,$s. Define the scaled variable $T=\dfrac{t}{3600}$ in hours. Calculate the values of $T$.

If variable $y$ ranges from $0$ to $25000$ mm and you define $Y = \frac{y}{1000}$ in metres, find the range of $Y$.

The exponential model is $y = 5e^{rx}.$ Show that plotting $\ln(y)$ versus $x$ yields the straight‐line equation $\ln(y) = \ln(5) + rx$.

Express the exponential model $y = 10e^{2x}$ as a linear relationship between $\ln(y)$ and $x$.

A company’s monthly revenue ranges from $200,000 to $1,500,000. If $R=\dfrac{\text{revenue}}{10^6}$ in millions of dollars, find the minimum and maximum values of $R$.

Convert the model $y = 5e^{rx}$ into a linear model involving $\log_{10}(y)$ and $x$.

Express the model $y = 12e^{-0.5x}$ in the form $\ln(y)=mx+c$ and identify $m$ and $c$.

An experiment yields the best‐fit line $\ln(P) = -0.2t + 3.5.$ Write down the exponential decay model for $P$ as a function of $t$.

A best‐fit line for $\ln(y)$ versus $x$ is given by $\ln(y) = 0.693 + 0.05x.$ Determine the parameters $A$ and $r$ of the equivalent exponential model $y = Ae^{rx}$.

Given the exponential decay model $N = 40e^{-0.5t},$ find the half‐life $T_{1/2}$ of $N$.

Given the straight‐line fit $\log_{10}(y) = 1.301 + 0.02x,$ find the equivalent exponential model $y = Ae^{rx}$.