Question

Q3. (a) Evaluate (3x+2)/(x+1)+(x-2)(b) If the demand function is p=25-3X-3X2"". Find the consumer's surplus when p=7​

Answer 1

SOLUTION

TO DETERMINE

$\displaystyle \: 1. \: \sf \: \frac{(3x + 2)}{(x + 1)} + (x - 2)$

2. If the demand function is p = 25 - 3x - 3x² . Find the consumer's surplus when p = 7

EVALUATION

1.

$\displaystyle \sf \: \frac{(3x + 2)}{(x + 1)} + (x - 2)$

$\displaystyle \sf \: = \frac{(3x + 2) + (x + 1)(x - 2)}{(x + 1)}$

$\displaystyle \sf \: = \frac{(3x + 2) + {x}^{2} - x - 2}{(x + 1)}$

$\displaystyle \sf \: = \frac{ {x}^{2} + 2x }{(x + 1)}$

2.

Here it is given that the demand function is p = 25 - 3x - 3x²

For p = 7 we get

25 - 3x - 3x² = 7

⇒ 3x² + 3x - 18 = 0

⇒ x² + x - 6 = 0

⇒ x = - 3 , x = 2

Since x can not be negative

So x = 2

Now the consumer's surplus when p = 7

$\displaystyle = \sf\int\limits_{0}^{x} p(x) \, dx - xp(x)$

$\displaystyle = \sf\int\limits_{0}^{2} (25 - 3x - 3 {x}^{2}) \, dx - 2p(2)$

$\displaystyle = \sf \left. 25x - 3\frac{x^2}{2} - {x}^{3} \right|_0^2 - 2p(2)$

$\displaystyle = \sf (25 \times 2) - \frac{3}{2} \times 4 - {2}^{3} - (2 \times 7)$

$\displaystyle = \sf 50 - 6 - 8 - 14$

$\displaystyle = \sf 22$

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