In the diagram below, two of the sides have a length of $s$, where $0 < s < 6$ and $s \in \mathbb{R}$. All angles are $90^{\circ}$ or $270^{\circ}$. (a) Find a formula (in algebra) for the area of this shape, in terms of $s$. (b) Show that (or explain why) the perimeter of this shape is always 32, no matter what the value of $s$.

Junior Cycle Mathematics ALGEBRA - Expressions: In the diagram below, two of the

In the diagram below, two of the sides have a length of $s$, where $0 < s < 6$ and $s \in \mathbb{R}$ - Junior Cycle Mathematics - Question 11 - 2019

Question 11

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In the diagram below, two of the sides have a length of $s$, where $0 < s < 6$ and $s \in \mathbb{R}$. All angles are $90^{\circ}$ or $270^{\circ}$. (a) Find a fo... show full transcript

Worked Solution & Example Answer:In the diagram below, two of the sides have a length of $s$, where $0 < s < 6$ and $s \in \mathbb{R}$ - Junior Cycle Mathematics - Question 11 - 2019

Step 1

Find a formula (in algebra) for the area of this shape, in terms of $s$.

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Answer

To find the area of the shape, we can split it into rectangles. The overall dimensions of the shape are 10 units in width and $(10 - s)$ in height due to the part where the length is $s$ .

The area can be calculated as follows:

The height is contributed by the full height minus the height where the length $s$ $s$ is present:
- Area = Width × Height
- Area = $10 \times (6) - s \times s$
- Area = $60 - s^2$

Thus, the formula for the area is: $ext{Area} = 60 - s^2$

Step 2

Show that (or explain why) the perimeter of this shape is always 32, no matter what the value of $s$.

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Answer

To find the perimeter, we need to sum the lengths of all sides. The perimeter can be calculated as follows:

The lengths are:

Left Side: $6$ units
Bottom Side: $10$ units
Right Side: $(6 - s)$ units
Top Side: $s$ units (upward)

Thus, the perimeter $P$ is given by: $P = 10 + 6 + (6 - s) + s$ This simplifies to: $P = 10 + 6 + 6 = 32$

As we can see, $s$ cancels out, meaning the perimeter remains constant at 32 units regardless of the value of $s$ .

In the diagram below, two of the sides have a length of $s$, where $0 < s < 6$ and $s \in \mathbb{R}$ - Junior Cycle Mathematics - Question 11 - 2019

Worked Solution & Example Answer:In the diagram below, two of the sides have a length of $s$, where $0 < s < 6$ and $s \in \mathbb{R}$ - Junior Cycle Mathematics - Question 11 - 2019

Find a formula (in algebra) for the area of this shape, in terms of $s$.

Show that (or explain why) the perimeter of this shape is always 32, no matter what the value of $s$.

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