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 m$^2$, (b) form and solve a quadratic inequality in $x$. (c) Hence state the range of the possible values of $x$.

Photo AI

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 $x > 1.7$

96%

114 rated

Answer

To find the perimeter of the garden, we can set up the equation based on the lengths provided in the diagram:

$P = 20x + 6$

From the question, we know that the perimeter must be greater than 40 meters:

$20x + 6 > 40$

Solving for $x$ :

$20x > 40 - 6$
$20x > 34$
$x > rac{34}{20}$
$x > 1.7$

Step 2

form and solve a quadratic inequality in $x$

99%

104 rated

Answer

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

$A = 2x(6x + 3) + 4x(4x) - 2x(4x + 3)$
$A = 2x(6x + 3) + 16x^2 - 2x(4x + 3)$
$A = 12x^2 + 6x + 16x^2 - 8x - 6x$
$A = 28x^2 - 6x$

Given that the area must be less than 120 m $^2$ :

$28x^2 - 6x < 120$
$28x^2 - 6x - 120 < 0$

Next, we solve the quadratic inequality:

Using the quadratic formula: $x = rac{-b ext{±} ext{sqrt}(b^2 - 4ac)}{2a}$ where $a = 28$ , $b = -6$ , and $c = -120$ .

Calculating the discriminant: $b^2 - 4ac = (-6)^2 - 4(28)(-120) = 36 + 13440 = 13476$

Finding the roots: $x = rac{6 ext{\pm} ext{sqrt}(13476)}{56}$ This gives us the critical values needed to determine the intervals for the quadratic inequality.

Step 3

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

96%

101 rated

Answer

After solving the inequality and evaluating the intervals, we express the final range of $x$ :

$1.7 < x < rac{5}{2}$
This implies that the permissible values for $x$ lie within the interval $(1.7, 2.5)$ , confirming both conditions provided in the question.

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 $x$

Join the A-Level students using SimpleStudy...