Tag: to find the maximum profit if marginal revenue and marginal cost function are given:

Questions Related to to find the maximum profit if marginal revenue and marginal cost function are given:

The cost function of a firm $C(x)=4x^2-x+70$. Find the average cost when $x=3$.

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. $\dfrac{104}{3}$

  2. $\dfrac{103}{3}$

  3. $\dfrac{105}{3}$

  4. $\dfrac{103}{4}$

Show answer
Correct Option: B
Explanation:

$\Rightarrow$   We have, $C(x)=4x^2-x+70$.

$\Rightarrow$   Average cost = $\dfrac{C(x)}{x}$
$\Rightarrow$   Average cost = $\dfrac{4x^2-x+70}{x}$
$\Rightarrow$   Average cost = $4x-1+\dfrac{70}{x}$
$\Rightarrow$   Substitute value of $x=3$.
$\Rightarrow$  Average cost = $12-1+\dfrac{70}{3}=\dfrac{36-3+70}{3}=\dfrac{103}{3}$
$\therefore$     $Average\, cost=\dfrac{103}{3}$

The demand function of a monopolist is given by $p=1500-2x-x^2$. Find the marginal revenue when $x=10$.

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. $1170$

  2. $1160$

  3. $1150$

  4. None of these

Show answer
Correct Option: B
Explanation:

$\Rightarrow$  We have, $p=1500-2x-x^2$

$\Rightarrow$  Revenue Function   $R=p\times x$
$\therefore$       $R=1500x-2x^2-x^3$.
$\Rightarrow$   Marginal revenue = $\dfrac{d}{dx}R$

$\Rightarrow$   Marginal revenue = $\dfrac{d}{dx}(1500x-2x^2-x^3)$ 

$\Rightarrow$   Marginal revenue = $1500-4x-3x^2$
$\Rightarrow$   Now, substitute $x=10$.
$\Rightarrow$   Marginal revenue = $1500-2(100)-3(100)^2=1160$
$\therefore$   Marginal revenue is $1160$.

Given the marginal cost function $\dfrac{2x}{3}+3-\dfrac{16}{x^2}$, find  average cost function.

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. $\dfrac{1}{3}x^2+3x-7+\dfrac{16}{x}$

  2. $\dfrac{1}{2}x^2+3x-7+\dfrac{16}{x}$

  3. $\dfrac{1}{4}x^2+3x-7+\dfrac{16}{x}$

  4. None of these

Show answer
Correct Option: A
Explanation:

$\Rightarrow$  We have $MC=\dfrac{2x}{3}+3-\dfrac{16}{x^2}$


$\Rightarrow$  $Average\,\,cost=\int (MC)dx$ 


$\Rightarrow$   $Average\,\,cost=\int (\dfrac{2x}{3}+3-\dfrac{16}{x^2})dx$


$\therefore$   $Average\,\,cost=\dfrac{2x^2}{2\times 3}+3x-7+\dfrac{16}{x}$

$\Rightarrow$   $Average\,\,cost=\dfrac{1}{3}x^2+3x-7+\dfrac{16}{x}$

The cost function of a firm $C(x)=3x^2-2x+3$. Find the average cost when $x=3$.

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. 8

  2. 9

  3. 10

  4. 12

Show answer
Correct Option: A
Explanation:

$\Rightarrow$  We have, $C(x)=3x^2-2x+3$

$\Rightarrow$  Average cost = $\dfrac{C(x)}{x}$

$\Rightarrow$  Average cost = $\dfrac{3x^2-2x+3}{x}$

$\Rightarrow$  Average cost = $3x-2+\dfrac{3}{x}$.

$\Rightarrow$  Substitute value of $x=3$,
$\Rightarrow$  Average cost = $3\times 3-2+\dfrac{3}{3}=9-2+1=8$
$\therefore$    $Average\, cost\,=8$. 

If the demanding Law is given by $q = \dfrac{20}{p+1}$, find the elasticity of demand with respect to price at the point when $p = 3.$

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. $\dfrac43$

  2. $-\dfrac34$

  3. $\dfrac23$

  4. $-\dfrac32$

Show answer
Correct Option: B
Explanation:
Elasticity of demand $=\cfrac{\cfrac{dq}{q}}{\cfrac{dp}{p}}=-\cfrac{p}{(p+1)}$
When $p=3$
Elasticity of demand $=-\cfrac{3}{4}$

If the total cost function for a manufacturer is given by $C =\dfrac{5x^2}{\sqrt(x^2+3)}+5000$, find marginal cost function.

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. $\dfrac{3x(x^2+6)}{(x^2+3)^{(3/2)}}$

  2. $\dfrac{4x(x^2+6)}{(x^2+3)^{(3/2)}}$

  3. $\dfrac{5x(x^2+6)}{(x^2+3)^{(3/2)}}$

  4. None of these

Show answer
Correct Option: C
Explanation:

Given cost function $C\left(x\right)=\dfrac{5x^2}{\sqrt{\left(x^2+3\right)}}+5000$


Marginal cost function is given by $C'\left(x\right)$

$C'\left(x\right)=\dfrac{\left(\sqrt{x^2+3}\right)\left(10x\right)-\dfrac{5x^2}{2\sqrt{x^2+3}}\times\left({2x+10}\right)}{\left(\sqrt{x^2+3}\right)^2}$

$\dfrac{d}{dx}\left(\dfrac{u}{v}\right)=\dfrac{v\dfrac{du}{dx}-u\dfrac{dv}{dx}}{v^2}$

$C'\left(x\right)=\dfrac{2\left(x^2+3\right)\left(10x\right)-5x^2\left(2x\right)}{2\left(\sqrt{x^2+3}\right)^3}=\dfrac{5x\left(x^2+6\right)}{\left(x^2+3\right)^\tfrac{3}{2}}$