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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`

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#### Solution

Total cost C = 40 + 2x and Price p = 120 − x

Profit π = R – C

∴ π = 120x – x^{2} – (40 + 2x)

= 120x – x^{2} – 40 – 2x

= **– x ^{2} + 118x – 40**

Differentiating w.r.t. x,

`("d"pi)/("d"x)` = **– 2x + 118**

Since Profit is increasing,

`("d"pi)/("d"x)` > 0

∴ – 2x + 118 > 0

∴ 2x < 118

∴x < 59

∴ Profit is increasing for **x < 59**.

#### RELATED QUESTIONS

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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.

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**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 ______.

**State whether the following statement is True or False: **

If the marginal revenue is 50 and the price is ₹ 75, then elasticity of demand is 4

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 elasticity of demand for price ₹ 80.

**Solution:** Total cost C = 40 + 2x and Price p = 120 – x

p = 120 – x

∴ x = 120 – p

Differentiating w.r.t. p,

`("d"x)/("dp")` = `square`

∴ Elasticity of demand is given by η = `- "P"/x*("d"x)/("dp")`

∴ η = `square`

When p = 80, then elasticity of demand η = `square`

**Complete the following activity to find MPC, MPS, APC and APS, if the expenditure E _{c} of a person with income I is given as:**

E_{c} = (0.0003)I^{2} + (0.075)I^{2}

when I = 1000

If elasticity of demand η = 0 then demand is ______.

If 0 < η < 1 then the demand is ______.