Suppose a plant has height 1.75 feet and it grows by 0.5 feet each month. (i) Find the height after 7 months. (ii) Make a table of values for varying from 0 to 10 months and show how the height, , increases every month. (iii) Find an expression that relates and , and explain why it represents linear growth.

Introduction to Linear Polynomials | Exercise 2.4

Question 1

Suppose a plant has height 1.75 feet and it grows by 0.5 feet each month.

(i) Find the height after 7 months.

(ii) Make a table of values for $t$ varying from 0 to 10 months and show how the height, $h$ , increases every month.

(iii) Find an expression that relates $h$ and $t$ , and explain why it represents linear growth.

Solution

Understand the Question

Initial height: $1.75\text{ feet}$ (at month $t = 0$ ).
Growth rate: $0.5\text{ feet}$ per month.
The total height $h$ after $t$ months is calculated as: $\text{Total Height} = \text{Initial Height} + (\text{Monthly Growth} \times t)$

(i) Find the height after 7 months.

Step 1 · Calculate Height after 7 Months

Given growth per month $= 0.5\text{ feet}$ and initial height $= 1.75\text{ feet}$ .

\begin{aligned} \text{Growth in 7 months} &= 0.5 \times 7 \\[0.6em] &= 3.5\text{ feet} \end{aligned}

\begin{aligned} \text{Height after 7 months} &= 1.75 + 3.5 \\[0.6em] &= 5.25\text{ feet} \end{aligned}

Answer

(i) $5.25\text{ feet}$

(ii) Make a table of values for $t$ varying from 0 to 10 months and show how the height, $h$ , increases every month.

Step 1 · Construct the Table of Values

Starting at $t = 0$ with $h = 1.75\text{ feet}$ , add $0.5\text{ feet}$ for each subsequent month:

\begin{array}{|c|c|} \hline t\text{ (months)} & h\text{ (feet)} \\ \hline 0 & 1.75 \\ \hline 1 & 2.25 \\ \hline 2 & 2.75 \\ \hline 3 & 3.25 \\ \hline 4 & 3.75 \\ \hline 5 & 4.25 \\ \hline 6 & 4.75 \\ \hline 7 & 5.25 \\ \hline 8 & 5.75 \\ \hline 9 & 6.25 \\ \hline 10 & 6.75 \\ \hline \end{array}

Answer

(ii) $$ \begin{array}{|c|c|} \hline t\text{ (months)} & h\text{ (feet)} \ \hline 0 & 1.75 \ \hline 1 & 2.25 \ \hline 2 & 2.75 \ \hline 3 & 3.25 \ \hline 4 & 3.75 \ \hline 5 & 4.25 \ \hline 6 & 4.75 \ \hline 7 & 5.25 \ \hline 8 & 5.75 \ \hline 9 & 6.25 \ \hline 10 & 6.75 \ \hline \end{array}

(iii) Find an expression that relates $h$ and $t$ , and explain why it represents linear growth.

Step 1 · Formulate Expression and Explain Linear Growth

Let $h$ be the height in feet and $t$ be the time in months.

h = 1.75 + 0.5t

This represents linear growth because:

The rate of change is constant ( $0.5\text{ feet}$ per month).
The variable $t$ is of degree $1$ in the equation.

Answer

(iii) $h = 1.75 + 0.5t$ ; it represents linear growth because the height increases at a constant rate of $0.5\text{ feet}$ each month.

Common Mistakes

Forgetting Initial Height: Calculating only the growth ( $0.5t$ ) and forgetting to add the initial height of $1.75\text{ feet}$ .
Off-by-One in Table Starting Value: Beginning the table at $t = 1$ with $1.75\text{ feet}$ instead of at $t = 0$ (the initial state).

Question 1

More questions in Exercise 2.4