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प्रश्न
The manufacturing company produces x items at the total cost of ₹ 180 + 4x. The demand function for this product is P = (240 − 𝑥). Find x for which profit is increasing
उत्तर
Now, Profit = Revenue − Total cost
∴ π = R − C
= 240x − x2 − (180 + 4x)
= 240x − x2 − 180 − 4x
∴ π = − x2 + 236x − 180
∴ `("d"pi)/("d"x)` = −2x + 236 = 2(− x + 118)
Since profit is an increasing function, `("d"pi)/("d"x)` > 0
∴ 2(−x + 118) > 0
∴ − x + 118 > 0
∴ 118 > x
∴ x < 118
∴ The profit is increasing for x < 118.
संबंधित प्रश्न
Find the marginal revenue if the average revenue is 45 and elasticity of demand is 5.
A manufacturing company produces x items at the total cost of Rs (180 + 4x). The demand function of this product is P = (240 − x). Find x for which profit is increasing.
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If the demand function is D = 50 – 3p – p2. Find the elasticity of demand at p = 5 comment on the result
If the demand function is D = 50 – 3p – p2. Find the elasticity of demand at p = 2 comment on the result
For the demand function D = 100 – `p^2/2`. Find the elasticity of demand at p = 10 and comment on the results.
A manufacturing company produces x items at a total cost of ₹ 40 + 2x. Their price is given as p = 120 – x. Find the value of x for which also find an elasticity of demand for price 80.
Fill in the blank:
A road of 108 m length is bent to form a rectangle. If the area of the rectangle is maximum, then its dimensions are _______.
If the marginal revenue is 28 and elasticity of demand is 3, then the price is ______.
A manufacturing company produces x items at a total cost of ₹ 40 + 2x. Their price per item is given as p = 120 – x. Find the value of x for which revenue is increasing
Solution: Total cost C = 40 + 2x and Price p = 120 – x
Revenue R = `square`
Differentiating w.r.t. x,
∴ `("dR")/("d"x) = square`
Since Revenue is increasing,
∴ `("dR")/("d"x)` > 0
∴ Revenue is increasing for `square`
A manufacturing company produces x items at a total cost of ₹ 40 + 2x. Their price per item is given as p = 120 – x. Find the value of x for which profit is increasing
Solution: Total cost C = 40 + 2x and Price p = 120 − x
Profit π = R – C
∴ π = `square`
Differentiating w.r.t. x,
`("d"pi)/("d"x)` = `square`
Since Profit is increasing,
`("d"pi)/("d"x)` > 0
∴ Profit is increasing for `square`
Complete the following activity to find MPC, MPS, APC and APS, if the expenditure Ec of a person with income I is given as:
Ec = (0.0003)I2 + (0.075)I2
when I = 1000
If elasticity of demand η = 0 then demand is ______.
If f(x) = x3 – 3x2 + 3x – 100, x ∈ R then f"(x) is ______.
If 0 < η < 1 then the demand is ______.
In a factory, for production of Q articles, standing charges are ₹500, labour charges are ₹700 and processing charges are 50Q. The price of an article is 1700 - 3Q. Complete the following activity to find the values of Q for which the profit is increasing.
Solution: Let C be the cost of production of Q articles.
Then C = standing charges + labour charges + processing charges
∴ C = `square`
Revenue R = P·Q = (1700 - 3Q)Q = 1700Q- 3Q2
Profit `pi = R - C = square`
Differentiating w.r.t. Q, we get
`(dpi)/(dQ) = square`
If profit is increasing , then `(dpi)/(dQ) >0`
∴ `Q < square`
Hence, profit is increasing for `Q < square`
