Respuesta :
the demand and supply are in equilibrium when the amount demanded is exactly the same amount as the amount supplied, or namely when demand = supply.
[tex] \bf \stackrel{equilibrium}{\stackrel{supply}{-0.1x^2-x+50}~~=~~\stackrel{demand}{0.1x^2+2x+30}}\qquad \implies 0=0.2x^2+3x-20\\\\\\\stackrel{\textit{multiplying both sides by 10}}{0=2x^2+30x-200}\implies \stackrel{\textit{dividing both sides by 2}}{0=x^2+15x-100}\\\\\\0=(x+20)(x-5)\implies \begin{array}{|c|ll}\cline{1-1}x\\\cline{1-1}\\-20\qquad \blacktriangleright 5 \blacktriangleleft\\\\\cline{1-1}\end{array} [/tex]
we can't use -20, because the amount cannot be a negative value.
At equilibrium, the function of demand and supply are equal
The equilibrium quantity where the demand and supply are equal is approximately 1,281 units tents
The reason the above value is correct is as follows:
The given function for the weekly demand for Sportsman 5 × 7 tents is presented as follows;
p = -0.1·x² - x + 50
The given function for the weekly supply for Sportsman 5 × 7 tents is presented as follows;
p = 0.1·x² - 2·x + 30
Where:
p is in dollars, and x is measured in units of a (one) hundred
Required:
To find the equilibrium quantity
Solution:
At equilibrium, the weekly demand and supply are equal, therefore, we have;
-0.1·x² - x + 50 = 0.1·x² - 2·x + 30
(0.1·x² - 2·x + 30) - (-0.1·x² - x + 50) = 0
0.2·x² - x - 20 = 0
Dividing by -0.2 gives;
x² - 5·x - 100 = 0
Which gives;
[tex]x = \dfrac{5 \pm\sqrt{(-5)^2 - 4 \times 1 \times (-100)} }{2 \times 1} = \dfrac{5 \pm 5 \times \sqrt{17} }{2 }[/tex]
x ≈ -7.81 or x ≈ 12.81
The positive value of x is used to give;
The equilibrium quantity, x ≈ 12.81 hundred units, which is approximately 1,281 units
The equilibrium quantity is approximately 1,281 units tents
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