Introduction to Linear Polynomials | EOT

Question 12

*12. Look at the first three stages of a growing pattern of hexagons made using matchsticks. A new hexagon gets added at every stage which shares a side with the last hexagon of the previous stage.

(i) Draw the next two stages of the pattern. How many matchsticks will be required at these stages?

(ii) Complete the following table.

(iii) Find a rule to determine the number of matchsticks required for the $n^{\text{th}}$ stage.

(iv) How many matchsticks will be required for the $15^{\text{th}}$ stage of the pattern?

(v) Can 200 matchsticks form a stage in this pattern? Justify your answer.

Solution

Understand the Question

At Stage 1, a single hexagon requires 6 matchsticks.
At each subsequent stage, an additional hexagon is attached which shares one common side with the previous hexagon, adding 5 matchsticks each time.
Therefore, the pattern of matchsticks forms an arithmetic sequence with first term $a = 6$ and common difference $d = 5$ : $\text{Number of matchsticks for stage } n = 6 + 5(n - 1) = 5n + 1$

(i) Draw the next two stages of the pattern. How many matchsticks will be required at these stages?

Step 1 · Determine Matchsticks for Stages 4 and 5

Each new stage adds $1$ hexagon and $5$ matchsticks to the previous stage:

Stage 4: $\text{Number of matchsticks} = 16 + 5 = 21$
Stage 5: $\text{Number of matchsticks} = 21 + 5 = 26$

Answer

(i) Stage 4 requires 21 matchsticks and Stage 5 requires 26 matchsticks.

(ii) Complete the following table.

Step 1 · Fill the Values for Stages 1 to 5

Using the matchstick counts for stages $1$ through $5$ :

\begin{array}{|c|c|c|c|c|c|} \hline \text{Stage } (n) & 1 & 2 & 3 & 4 & 5 \\ \hline \text{Number of matchsticks} & 6 & 11 & 16 & 21 & 26 \\ \hline \end{array}

Answer

(ii) $$ \begin{array}{|c|c|c|c|c|c|} \hline \text{Stage } (n) & 1 & 2 & 3 & 4 & 5 \ \hline \text{Number of matchsticks} & 6 & 11 & 16 & 21 & 26 \ \hline \end{array}

(iii) Find a rule to determine the number of matchsticks required for the $n^{\text{th}}$ stage.

Step 1 · Derive the General Formula

The first stage has $6$ matchsticks, and each of the remaining $(n - 1)$ stages adds $5$ matchsticks:

\begin{aligned} \text{Number of matchsticks} &= 6 + (n - 1) \times 5 \\ &= 6 + 5n - 5 \\ &= 5n + 1 \end{aligned}

Answer

(iii) $5n + 1$

(iv) How many matchsticks will be required for the $15^{\text{th}}$ stage of the pattern?

Step 1 · Calculate for $n = 15$

Substitute $n = 15$ into the rule $5n + 1$ :

\begin{aligned} \text{Number of matchsticks} &= 5(15) + 1 \\ &= 75 + 1 \\ &= 76 \end{aligned}

Answer

(iv) $76$

(v) Can 200 matchsticks form a stage in this pattern? Justify your answer.

Step 1 · Check if $n$ is a Whole Number

Set the formula $5n + 1$ equal to $200$ :

\begin{aligned} 5n + 1 &= 200 \\ 5n &= 199 \\ n &= \dfrac{199}{5} \\ n &= 39.8 \end{aligned}

Since the stage number $n$ must be a positive integer, $200$ matchsticks cannot form a stage in this pattern.

Answer

(v) No, because $n = \dfrac{199}{5} = 39.8$ is not an integer.

Common Mistakes

Multiplying by 6 for each stage: Incorrectly calculating the matchsticks as $6 \times n$ (which gives $6, 12, 18, \dots$ ), forgetting that adjacent hexagons share a side and only add $5$ matchsticks per stage.
Formula Offset Error: Writing the formula as $5n$ or $6n - 1$ instead of $5n + 1$ .

More questions in EOT

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(i) 6 months

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Express this as a linear pattern.

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(i) $y = -3x + 4$

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(iv) $3y = 6x - 11$

Are any of the lines parallel?

If the temperature of a liquid can be measured in Kelvin units as $x \text{ K}$ and in Fahrenheit units as $y^{\circ}\text{F}$ , the relation between the two systems of measurement of temperature is given by the linear equation $y = \dfrac{9}{5}(x - 273) + 32$ .

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The work done by a body on the application of a constant force is the product of the constant force and the distance travelled by the body in the direction of the force. Express this in the form of a linear equation in two variables (work $w$ and distance $d$ ), and draw its graph by taking the constant force as 3 units. What is the work done when the distance travelled is 2 units? Verify it by plotting it on the graph.

Q10

*10. The graph of a linear polynomial $p(x)$ passes through the points $(1, 5)$ and $(3, 11)$ .

(i) Find the polynomial $p(x)$ .

(ii) Find the coordinates where the graph of $p(x)$ cuts the axes.

(iii) Draw the graph of $p(x)$ and verify your answers.

Q11

*11. Let $p(x) = ax + b$ and $q(x) = cx + d$ be two linear polynomials such that:

(i) $p(0) = 5$ .

(ii) The polynomial $p(x) - q(x)$ cuts the $x$ -axis at $(3, 0)$ .

(iii) The sum $p(x) + q(x)$ is equal to $6x + 4$ for all real $x$ .

Find the polynomials $p(x)$ and $q(x)$ .

Q12

*12. Look at the first three stages of a growing pattern of hexagons made using matchsticks. A new hexagon gets added at every stage which shares a side with the last hexagon of the previous stage.

(i) Draw the next two stages of the pattern. How many matchsticks will be required at these stages?

(ii) Complete the following table.

(iii) Find a rule to determine the number of matchsticks required for the $n^{\text{th}}$ stage.

(iv) How many matchsticks will be required for the $15^{\text{th}}$ stage of the pattern?

(v) Can 200 matchsticks form a stage in this pattern? Justify your answer.

Q13

*13. Let $p(x) = ax + b$ and $q(x) = cx + d$ be two linear polynomials such that:

(i) The graph of $p(x)$ passes through the points $(2, 3)$ and $(6, 11)$ .

(ii) The graph of $q(x)$ passes through the point $(4, -1)$ .

(iii) The graph of $q(x)$ is parallel to the graph of $p(x)$ .

Find the polynomials $p(x)$ and $q(x)$ . Also, find the coordinates of the point where these lines meet the $x$ -axis.

Q14

*14. What do all linear functions of the form $f(x) = ax + a, a > 0$ , have in common?