Advertisements
Advertisements
Question
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.
Advertisements
Solution
Total cost function (C) = 180 + 4x
Demand function (P) = 240 − x
Where x is the number of items produced.
Total revenue (R) = P × D
∴ R = x (240 − x)
∴ R = 240x − x2
Profit function π = R − C
∴ π = (240x − x2) − (180 + 4x)
∴ π = 240x − x2 − 180 − 4x
∴ π = − x2 + 236x − 180
Differentiating w.r.t.x,
∴ `"dπ"/"dx"` = − 2x + 236
Profit π is increasing if `"dπ"/"dx"` > 0
i.e. if − 2x + 236 > 0
i.e. if 236 > 2x
i.e. if x < `236/2`
i.e. if x < 118
∴ The profit is increasing for x < 118.
APPEARS IN
RELATED QUESTIONS
The demand function of a commodity at price P is given as, D = `40 - "5P"/8`. Check whether it is increasing or decreasing function.
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 revenue is increasing
The total cost of manufacturing x articles C = 47x + 300x2 – x4 . Find x, for which average cost is decreasing
Find the price, if the marginal revenue is 28 and elasticity of demand is 3.
If the demand function is D = `((p + 6)/(p − 3))`, find the elasticity of demand at p = 4.
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 = 6 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 revenue is increasing.
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 profit is increasing.
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 ______.
If the elasticity of demand η = 1, then demand is ______.
If 0 < η < 1, then the demand is ______.
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
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 f(x) = x3 – 3x2 + 3x – 100, x ∈ R then f"(x) is ______.
If 0 < η < 1 then the demand is ______.
