Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01R) | Year 2014 | June | Q#6

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Question

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.

Solution

a.
 

We are given that;

Perimeter of the garden, from the diagram, can be written as;

Therefore;

b.    

We are given that;

Area of the garden, from the diagram given below, can be written.

Expression for perimeter of a rectangle with width  and length   is:

Therefore;

Therefore;

We are required to solve the inequality;

We solve the following equation to find critical values of ;

Now we have two options;

Hence the critical points on the curve for the given condition are  & -3.

Standard form of quadratic equation is;

The graph of quadratic equation is a parabola. If  (‘a’ is positive) then parabola opens upwards  and its vertex is the minimum point on the graph.
If
 (‘a’ is negative) then parabola opens downwards and its vertex is the maximum point on the  graph.

We recognize that  it is an upwards opening parabola.

Therefore conditions for  are;

Hence;

c.    

Since we are dealing with a garden perimeter and area for which we have got;

 

For both inequalities to be true the overlapping period of two sets of value of x is what will make   them so;

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