Question

The diagram shows a parallelogram.

The area of the parallelogram is greater
than 10.5 cm

a) Show that 2x2 - 25x + 33 < 0
b) Find the range of possible values of x.
(3)
Note: Write your final answer in the format ... <x<...
​

Answer 1

Answer:

3/2 < x < 11

Step-by-step explanation:

Answer 2

Let us name the parallelogram ABCD in cyclic order from the leftmost vertex. So, as provided in the question :

Length AD = (12 - x) cm

Length CD = ( 2x - 1 ) cm

Angle C = 150 degrees

Let us draw a perpendicular from vertex A on the base CD touching it at point E. Thus, the triangle AED formed is right-angled at E.

a) The equation $2x^{2}-25x +33 < 0$ is satisfied by calculating the parallelogram's area.

As the sum of adjacent vertices of a parallelogram sum up to 180 degrees,

           ∠C + ∠D = 180°

           ∠D = 180° - ∠C

                 = 180° - 150°

                 = 30°

Applying the rules of trigonometry in triangle AED :

                   $sinD = AE/AD\\sin(30) = AE/(12-x)\\AE = (1/2)(12-x)\\AE = 6 - (x/2)$

The Area of the given parallelogram ABCD

          Area = (Base of the parallelogram i.e. CD) * ( height i.e. AE )

                   $= CD * AE\\= (2x-1)(6-(x/2))$

            $10.5 < (2x-1)(6-(x/2))$

            $10.5 < 12x- x^{2}-6 +(x/2)$

            $10.5 < (1/2)(24x- 2x^{2}-12 +x)$

            $21 < 25x- 2x^{2}-12$

            $2x^{2}-25x < -12 - 21$

            $2x^{2}-25x +33 < 0$

Hence, proved that given the area of the parallelogram is greater than 10.5 sq. units, the equation $2x^{2}-25x +33 < 0$ is true.

b) The possoble range for value pf "x" is between (3/2) and 11 i.e.

" (3/2) < x < 11 ".

The equation above is :

              $2x^{2}-25x +33 < 0$

              $2x^{2}-22x -3x+33 < 0$

              $2x(x-11) -3(x-11) < 0$

              $(2x-3)(x-11) < 0$

Equation 1 : The first term is positive

$2x-3 > 0\\2x > 3\\x > 3/2$

Equation 2 : The second factor is negative for the product to be negative

$x-11 < 0\\x < 11$

Hence, the range of possible values of x is 3/2 < x < 11.

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