A rectangular room has a width of $r$ m. The length of the room is 4 m longer than its width. Given that the perimeter of the room is greater than 19.2 m, a) show that $r > 2.8$. Given also that the area of the room is less than 21 m², (b) (i) write down an inequality, in terms of $r$, for the area of the room. (ii) Solve this inequality. (c) Hence find the range of possible values for $r$.

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A rectangular room has a width of $r$ m - Edexcel - A-Level Maths Pure - Question 9 - 2013 - Paper 2

Question 9

A rectangular room has a width of $r$ m. The length of the room is 4 m longer than its width. Given that the perimeter of the room is greater than 19.2 m, a) show t... show full transcript

Worked Solution & Example Answer:A rectangular room has a width of $r$ m - Edexcel - A-Level Maths Pure - Question 9 - 2013 - Paper 2

Step 1

a) show that $r > 2.8$

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Answer

To find the perimeter, we can use the formula:

$P = 2 \times (\text{length} + \text{width})$

Given that the width is $r$ and the length is $r + 4$ , the perimeter becomes:

$P = 2 \times (r + (r + 4)) = 2 \times (2r + 4) = 4r + 8$

We know that this perimeter is greater than 19.2 m:

$4r + 8 > 19.2$

First, we can subtract 8 from both sides:

$4r > 11.2$

Next, divide both sides by 4:

$r > 2.8$

Thus, we have shown that $r > 2.8$ .

Step 2

b) (i) write down an inequality, in terms of $r$, for the area of the room.

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Answer

The area of the room can be expressed as:

$A = \text{length} \times \text{width} = (r + 4) \times r = r^2 + 4r$

Given that the area is less than 21 m², we can write the inequality as:

$r^2 + 4r < 21$

Step 3

b) (ii) Solve this inequality.

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Answer

To solve the inequality:

$r^2 + 4r - 21 < 0$

We need to factor the quadratic:

$(r - 3)(r + 7) < 0$

Next, we find the critical points by setting the factored form to zero:

\ r + 7 = 0 \Rightarrow r = -7 $$ We can test intervals between the critical points (-7, 3): 1. For $r < -7$ (e.g., $r = -8$), $( -8 - 3)(-8 + 7) > 0 $ 2. For $-7 < r < 3$ (e.g., $r = 0$), $(0 - 3)(0 + 7) < 0 $ 3. For $r > 3$ (e.g., $r = 4$), $(4 - 3)(4 + 7) > 0 $ The solution to the inequality is: $$ -7 < r < 3 $$

Step 4

c) Hence find the range of possible values for $r$.

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Answer

From part (a), we have shown that $r > 2.8$ . From part (b), we found that $-7 < r < 3$ .

Combining these results, the range of possible values for $r$ is:

$2.8 < r < 3$

A rectangular room has a width of $r$ m - Edexcel - A-Level Maths Pure - Question 9 - 2013 - Paper 2

Worked Solution & Example Answer:A rectangular room has a width of $r$ m - Edexcel - A-Level Maths Pure - Question 9 - 2013 - Paper 2

a) show that $r > 2.8$

b) (i) write down an inequality, in terms of $r$, for the area of the room.

b) (ii) Solve this inequality.

c) Hence find the range of possible values for $r$.

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