Question Type 1: Scaling data appropriately under the context Exercises

If variable $y$ ranges from $0$ to $25\,000\text{ mm}$ and you define $Y = \frac{y}{1000}$ in metres, find the range of $Y$.

An experiment yields the best-fit line

Express $P$ as an exponential decay model in the form $P = Ae^{rt}$, where $A$ and $r$ are constants.

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

A best‐fit line for $\ln(y)$ versus $x$ is given by

Determine the parameters $A$ and $r$ of the equivalent exponential model $y = Ae^{rx}$. [3 marks]

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

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

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

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

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

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