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ArbitrLikvidat [17]
3 years ago
14

A fried chicken franchise finds that the demand equation for its new roast chicken product, "Roasted Rooster," is given by p = 4

5 / q 1.5
where p is the price (in dollars) per quarter-chicken serving and q is the number of quarter-chicken servings that can be sold per hour at this price.

1) Express q as a function of p.

2) Find the price elasticity of demand when the price is set at $4.00 per serving.
Mathematics
1 answer:
LUCKY_DIMON [66]3 years ago
3 0

Answer:

1. q=(\dfrac{45}{p})^{\frac{2}{3}}

2. E_d=-\dfrac{2}{3}

Step-by-step explanation:

The given demand equation is

p=\dfrac{45}{q^{1.5}}

where p is the price (in dollars) per quarter-chicken serving and q is the number of quarter-chicken servings that can be sold per hour at this price.

Part 1 :

We need to Express q as a function of p.

The given equation can be rewritten as

q^{1.5}=\dfrac{45}{p}

Using the properties of exponent, we get

q=(\dfrac{45}{p})^{\frac{1}{1.5}}      [\because x^n=a\Rightarrow x=a^{\frac{1}{n}}]

q=(\dfrac{45}{p})^{\frac{2}{3}}

Therefore, the required equation is q=(\dfrac{45}{p})^{\frac{2}{3}}.

Part 2 :

q=(45)^{\frac{2}{3}}p^{-\frac{2}{3}}

Differentiate q with respect to p.

\dfrac{dq}{dp}=(45)^{\frac{2}{3}}(-\dfrac{2}{3})(p^{-\frac{2}{3}-1}})

\dfrac{dq}{dp}=(45)^{\frac{2}{3}}(-\dfrac{2}{3})(p^{-\frac{5}{3}})

\dfrac{dq}{dp}=(45)^{\frac{2}{3}}(-\dfrac{2}{3})(\dfrac{1}{p^{\frac{5}{3}}})

Formula for price elasticity of demand is

E_d=\dfrac{dq}{dp}\times \dfrac{p}{q}

E_d=(45)^{\frac{2}{3}}(-\dfrac{2}{3})(\dfrac{1}{p^{\frac{5}{3}}})\times \dfrac{p}{(45)^{\frac{2}{3}}p^{-\frac{2}{3}}}

Cancel out common factors.

E_d=(-\dfrac{2}{3})(\dfrac{1}{p^{\frac{5}{3}}})\times \dfrac{p}{p^{-\frac{2}{3}}}

Using the properties of exponents we get

E_d=-\dfrac{2}{3}(p^{-\frac{5}{3}+1-(-\frac{2}{3})})

E_d=-\dfrac{2}{3}(p^{0})

E_d=-\dfrac{2}{3}

Therefore, the price elasticity of demand is -2/3.

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