Respuesta :
Answer:
See expla below
Step-by-step explanation:
Given the demand function:
q = D (x) = 943 - 17 x
a) Find the elasticity:
Find the derivative of the demand function.
D'(x)= -17
Thus, elasticity expression is:
[tex]\frac{x D'(x)}{D'(x)}[/tex]
[tex] = \frac{x (-17)}{943 - 17x}[/tex]
[tex] = \frac{17x}{943 - 17x}[/tex]
Elasticity expression = [tex] E(x) = \frac{17x}{943 - 17x}[/tex]
b) At what price is the elasticity of demand equal to 1?
This means E(x) = 1
Substitute 1 for E(x) in the elasticity equation:
[tex] E(x) = \frac{17x}{943 - 17x}[/tex]
[tex] 1 = \frac{17x}{943 - 17x}[/tex]
Cross multiply:
[tex] 943 - 17x = 17x [/tex]
Collect like terms
[tex] 17x + 17x = 943 [/tex]
[tex] 34x = 943 [/tex]
[tex] x = \frac{943}{34} [/tex]
x = 27.74
Elasticity at the price of demand = 1 is 27.74
c) At what prices is the elasticity of demand elastic?
This means E(x) > 1
Therefore,
[tex] \frac{17x}{943 - 17x} > 1[/tex]
[tex] \frac{17x}{943 - 17x} > 1 [/tex]
Cross multiply:
[tex] 17x > 943 - 17x [/tex]
Collect like terms
[tex] 17x + 17x > 943 [/tex]
[tex] 34x > 943 [/tex]
[tex] x > \frac{943}{34} [/tex]
x > 27.74
The elasticity of demand is elastic at x > 27.74
d) At what prices is the elasticity of demand inelastic?
This means E(x) < 1
Therefore,
[tex] \frac{17x}{943 - 17x} < 1[/tex]
[tex] \frac{17x}{943 - 17x} < 1 [/tex]
Cross multiply:
[tex] 17x < 943 - 17x [/tex]
Collect like terms
[tex] 17x + 17x < 943 [/tex]
[tex] 34x < 943 [/tex]
[tex] x < \frac{943}{34} [/tex]
x < 27.74
The elasticity of demand is inelastic at x < 27.74
e) At what price is the revenue a maximum:
Total revenue will be:
R(x) = x D(x)
= x (943 - 17x)
= 943x - 17x²
R(x) = 934 - 17x(price that maximizes total revenue)
Take R(x) = 0
Thus,
0 = 943 - 17x
17x = 943x
[tex] x = \frac{943}{17} [/tex]
[tex] x = 27.74 [/tex]
Total revenue is maximun at x= 27.74 per cookie
f) At x = 21 per cookie, find the price:
Thus,
R (21) = (943 * 21) - (17 * 21²)
= 19803 - 7497
= 12306
At x = 27.74, find the price:
R(27.74) = (943 * 27.74) - (17 - 27.74²)
= 26158.82 - 13081.63
= 13077.19
We can see the new price of cookie causes the total revenue to decrease.
Therefore, with a small increase in price the total revenue will decrease.
The demand is elastic when the change in the quantity demanded with
respect to change in price is significant.
- [tex]A) \ \displaystyle \mathrm{Elasticity, \ } \underline{ \epsilon = \displaystyle \frac{17 \cdot x}{943 - 17\cdot x}}[/tex]
- B) The price at which the elasticity is one is approximately 27.74c
- C) The elasticity is elastic when the price is larger than 27.74c
- D) The elasticity is inelastic when the price is lesser than 27.74c
- E) The revenue is maximum when the price is approximately 27.74c
- F) 1. Increases
Reasons:
A) The demand function is; q = D(x) = 943 - 17·x
Where;
q = The quantity of cookies sold
x = The price of cookies
Elasticity of demand is given by the formula;
[tex]\displaystyle \epsilon = \mathbf{\frac{\Delta Q}{\Delta P} \times \frac{P}{Q}}[/tex]
[tex]\displaystyle \frac{\Delta Q}{\Delta P} = 17[/tex]
[tex]\displaystyle \frac{P}{Q} = \frac{x}{943 - 17\cdot x}[/tex]
Therefore;
[tex]\displaystyle \underline{ \mathrm{Elasticity, \ } \epsilon = \displaystyle \frac{17 \cdot x}{943 - 17\cdot x}}[/tex]
B) When the elasticity, ε = 1, we have;
[tex]\displaystyle \displaystyle \frac{17 \cdot x}{943 - 17\cdot x} = 1[/tex]
943 - 17·x = 17·x
943 = 17·x + 17·x = 34·x
[tex]\displaystyle x = \frac{943}{34} \approx 27.74[/tex]
When the elasticity, ε = 1, the price, x ≈ 27.74
C) The demand is elastic when, ε > 1, which gives;
[tex]\displaystyle \displaystyle \epsilon = \frac{17 \cdot x}{943 - 17\cdot x} > 1[/tex]
17·x > 943 - 17·x
[tex]\displaystyle x > \frac{943}{34}[/tex]
[tex]\displaystyle \frac{943}{34} \approx 27.74[/tex]
Which gives;
x > 27.74
The price at which the elasticity of demand is elastic is x > 27.74
D) The elasticity of demand is inelastic when we have;
[tex]\displaystyle \displaystyle \epsilon = \mathbf{ \frac{17 \cdot x}{943 - 17\cdot x} }< 1[/tex]
Which gives;
17·x < 943 - 17·x
17·x + 17·x < 943
34·x < 943
[tex]\displaystyle x < \frac{943}{34}[/tex]
x < 27.74
E) The total revenue, R = Price × Quantity sold
∴ R = x × 943 - 17·x = 943·x - 17·x²
When the revenue is maximum, we have;
[tex]\displaystyle R' = 0 = \frac{d}{dx} \left(943 \cdot x - 17 \cdot x^2 \right) = 943 - 34 \cdot x[/tex]
Which gives;
943 - 34·x = 0
943 = 34·x
x ≈ 27.74
When the revenue is maximum, the price, x ≈ 27.74
F) When the price is 21c
The total revenue is given from the revenue formula and the price at maximum revenue as follows;
The function for the total revenue is, R = 943·x - 17·x²
The sign of the leading coefficient is negative, the graph of the total revenue function is concave down and from question e, the price at the maximum revenue, x = 27.74, which gives;
At points before x = 27.74, a small increase in price will cause the total revenue to increase
Therefore, at x = 21c, a small increase in price will cause the total revenue to increase. The correct option is 1. Increase
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https://brainly.com/question/15654343
