The consumer demand equation for tissues is given by
q = (97 − p)2,
where p is the price per case of tissues and q is the demand in weekly sales.
(a) Determine the price elasticity of demand E when the price is set at $26. (Round your answer to three decimal places.)
E =______
Interpret your answer.
The demand is going down by _____ % per 1% increase in price at that price level.
(b) At what price should tissues be sold in order to maximize the revenue? (Round your answer to the nearest cent.)
$____.
(c) Approximately how many cases of tissues would be demanded at that price? (Round your answer to the nearest whole number.)_______ cases per week
Honestly of Dec 6, 2016, why are you so bored that you go onto homework help websites and criticize people looking for help?
(a) To determine the price elasticity of demand (E) when the price is set at $26, we need to take the derivative of the demand equation with respect to price and then substitute the given price value.
The demand equation is:
q = (97 - p)^2
Taking the derivative of q with respect to p:
dq/dp = -2(97 - p)
Now we substitute p = 26 into the derivative:
dq/dp = -2(97 - 26) = -2(71) = -142
To find the price elasticity of demand, we divide the derivative by the price and take the absolute value:
E = |dq/dp * (p/q)| = |-142 * (26/((97 - 26)^2))|
Calculating:
E = |-142 * (26/((97 - 26)^2))| ≈ 3.977
Therefore, the price elasticity of demand when the price is $26 is approximately 3.977.
Interpretation:
The price elasticity of demand measures the responsiveness of the quantity demanded to a change in price. In this case, a price elasticity of demand of 3.977 implies that for every 1% increase in price at the given price level, the demand for tissues decreases by approximately 3.977%.
(b) To maximize revenue, we need to find the price that corresponds to the maximum point on the demand curve. In this case, we can find it by taking the derivative of the demand equation and setting it equal to zero.
The demand equation is:
q = (97 - p)^2
Taking the derivative of q with respect to p:
dq/dp = -2(97 - p)
Setting dq/dp = 0:
-2(97 - p) = 0
97 - p = 0
p = 97
Therefore, tissues should be sold at a price of $97 to maximize revenue.
(c) To find the number of cases of tissues demanded at the price of $97, we can substitute this price into the demand equation:
q = (97 - p)^2
q = (97 - 97)^2
q = 0^2
q = 0
Therefore, approximately 0 cases of tissues would be demanded at a price of $97 per week.
To determine the price elasticity of demand (E) at a given price, we can use the formula:
E = -dq/dp * (p/q),
where dq/dp represents the derivative of the demand equation with respect to price, and (p/q) represents the ratio of price to quantity.
In this case, the demand equation is given by q = (97 − p)^2.
(a) To find the price elasticity of demand when the price is $26, we need to first calculate dq/dp and then evaluate the expression at that particular price.
Taking the derivative of the demand equation with respect to price, we get:
dq/dp = 2(97 - p) * (-1) = -2(97 - p).
Now, substitute p = $26 into the equation:
dq/dp = -2(97 - 26) = -2(71) = -142.
Next, calculate the ratio (p/q) at the given price:
(p/q) = (26/[(97 - 26)^2]) = (26/71^2).
Finally, multiply the two values together to find the price elasticity of demand:
E = -dq/dp * (p/q) = -142 * (26/71^2).
By evaluating this expression, we can determine the price elasticity of demand, E.
(b) To find the price at which tissues should be sold to maximize revenue, we need to determine the price at which the demand is most elastic. This occurs when the price elasticity of demand (E) equals -1. In other words, we want to find the price (p) such that E = -1.
Using the formula for E derived above, we can solve for p.
(c) To find the approximate number of cases demanded at the price that maximizes revenue, substitute the price (p) from part (b) into the demand equation q = (97 − p)^2. This will give us the corresponding quantity (q).
By following these steps, we can determine the answers to parts (a), (b), and (c) of the problem.