Question

(i) A rectangular tile has length 4x cm and width (x + 3) cm. The area of the rectangle is less than 112 cm². By writing down and solving an inequality, determine the set of possible values of x. [6] (ii) A second rectangular tile of length 4ycm and width (y + 3) cm has a rectangle of length 2ycm and width ycm removed from one corner as shown in the diagram. y+3 4y 2y Given that the perimeter of this tile is between 20 cm and 54 cm, determine the set of possible values [5] of y.​

Answer 1

(i)
`\(x > 4\) for \(4x * (x + 3) < 112\).` (ii)
`\((7)/(4) \leq y \leq 6\) for \(20 \leq 8y + 6 \leq 54\).`

(i) Let's denote the length of the rectangular tile as L and the width as W. According to the given information:

`\[L = 4x \ \text{cm}\]`

`\[W = (x + 3) \ \text{cm}\]`

The area of the rectangle is given by the formula:
`\[ \text{Area} = L * W \]`

We are given that the area is less than 112 cm²:

`\[4x * (x + 3) < 112\]`

Now, we can solve this inequality:

`\[4x^2 + 12x < 112\]`

`\[4x^2 + 12x - 112 < 0\]`

Now, factorizing the quadratic expression:

`\[(2x - 8)(2x + 14) < 0\]`

This inequality holds true when
`\(2x - 8 > 0\)` and
`\(2x + 14 < 0\)`. Solving these inequalities separately, we get
`\(x > 4\) and \(x < -7\).` However, since the length and width of the tile cannot be negative, we discard
`\(x < -7\)` and conclude that the set of possible values for
`\(x\)` is
`\(x > 4\)`.

(ii) The perimeter of the second rectangular tile is given by the sum of the lengths of its four sides:

`\[P = 2(4y) + 2(y + 3) - 2y\]`

Simplifying this expression, we get:

`\[P = 8y + 2y + 6 - 2y = 8y + 6\]`

We are given that the perimeter is between 20 cm and 54 cm:

`\[20 \leq 8y + 6 \leq 54\]`

Solving for
`\(y\):`

`\[14 \leq 8y \leq 48\]`

`\[ (7)/(4) \leq y \leq 6\]`