Figure 1 shows the plan of a garden. The marked angles are right angles. The six edges are straight lines. The lengths shown in the diagram are given in metres. Given that the perimeter of the garden is greater than 40 m, (a) show that $x > 1.7$. Given that the area of the garden is less than 120 m², (b) form and solve a quadratic inequality in $x$. (c) Hence state the range of the possible values of $x$.

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Figure 1 shows the plan of a garden - Edexcel - A-Level Maths Pure - Question 8 - 2014 - Paper 2

Question 8

Figure 1 shows the plan of a garden. The marked angles are right angles. The six edges are straight lines. The lengths shown in the diagram are given in metres. G... show full transcript

Worked Solution & Example Answer:Figure 1 shows the plan of a garden - Edexcel - A-Level Maths Pure - Question 8 - 2014 - Paper 2

Step 1

show that $x > 1.7$

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Answer

To find the perimeter $P$ of the garden, we sum all the sides as given in the diagram:

$P = 2(2x + 1) + 2(2x) + 4x + (6x + 3) + 2$

This simplifies to:

$P = 20x + 6$

We are given that the perimeter is greater than 40 m, so:

$20x + 6 > 40$

Subtracting 6 from both sides:

$20x > 34$

Dividing both sides by 20 gives:

$x > 1.7$

Step 2

form and solve a quadratic inequality in $x$

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Answer

The area $A$ of the garden can be expressed as:

$A = 2(2x + 1) + 2x(4x)$

Simplifying this gives:

$A = 16x^2 + 8x$

Now, we set up the inequality based on the given constraints:

$16x^2 + 8x < 120$

Rearranging this leads to:

$16x^2 + 8x - 120 < 0$

Dividing everything by 8 results in:

$2x^2 + x - 15 < 0$

Next, we factor this expression:

$(2x - 5)(x + 3) < 0$

To solve this, we find critical values by setting each factor equal to zero:

ightarrow x = 2.5$$ $$x + 3 = 0 ightarrow x = -3$$ Now we test intervals around the critical points (-3 and 2.5) to find the regions where the inequality holds.

Step 3

Hence state the range of the possible values of $x$

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Answer

The solution to the quadratic inequality $(2x - 5)(x + 3) < 0$ indicates that:

It is negative between the roots, i.e., for $-3 < x < 2.5$ .

Combining this with the result from part (a), where we found that $x > 1.7$ , the combined inequality is:

$1.7 < x < 2.5$

This gives the final range of possible values for $x$ .

Figure 1 shows the plan of a garden - Edexcel - A-Level Maths Pure - Question 8 - 2014 - Paper 2

Worked Solution & Example Answer:Figure 1 shows the plan of a garden - Edexcel - A-Level Maths Pure - Question 8 - 2014 - Paper 2

show that $x > 1.7$

form and solve a quadratic inequality in $x$

Hence state the range of the possible values of $x$

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