Tag: applications of calculus

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Questions Related to applications of calculus

Multiple choice business mathematics and statistics applications of calculus revenue functions from marginal revenue functions minimization of cost function and maximization of revenue function and profit function integral calculus – ii
A firm $ABC$ starts producing pens and finds that the production cost of each pen is Rs $10$, and the fixed expenditures of production is Rs. $4500$. If each pen is sold for Rs. $25$, find break-even point.
  1. $x=200$
  2. $x=100$
  3. $x=400$
  4. $x=300$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Manufacturing cost $=10$
Number of pens $=10$
Total manufacturing cost $=10x+4500$
Selling cost $=25x$
Break even value $\Rightarrow $ manufacturing cost $=$ Selling cost 
$\Rightarrow 10x+4500=25x$
$x=300$
Multiple choice business mathematics and statistics applications of calculus revenue functions from marginal revenue functions minimization of cost function and maximization of revenue function and profit function integral calculus – ii
A company sells its product at the rate of Rs. $6$ per unit. The variable costs are estimated to run $25\%$ of the total revenue received. If the fixed costs for the product are Rs. $4500$. Find the total cost function.
  1. $4500x$
  2. $4500x+\dfrac{1}{2}x$
  3. $4500x+\dfrac{3}{2}x$
  4. $4500x+\dfrac{5}{2}x$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\Rightarrow$  Here, price per unit $(p)=Rs.6$ and fix cost is $Rs.4500$.

$\Rightarrow$  $Total\,\, revenue\,\,R(x)=p.x=6x$  where $x$ is number of unit sold.
$\Rightarrow$  $Total\, \,cost\,\, function\,\,C(x)=4500x+\dfrac{25}{100}R(x)$
$\Rightarrow$  $C(x)=4500x+\dfrac{25}{100}\times 6x$

$\Rightarrow$  $C(x)=4500x+\dfrac{3}{2}x$ 

Multiple choice business mathematics and statistics applications of calculus revenue functions from marginal revenue functions minimization of cost function and maximization of revenue function and profit function integral calculus – ii
A company sells its product at the rate of Rs. $6$ per unit. The variable costs are estimated to run $25\%$ of the total revenue received. If the fixed costs for the product are Rs. $4500$. Find the profit function.
  1. $6x-\dfrac{3}{2}x-4500$
  2. $6x-\dfrac{1}{2}x-4500$
  3. $6x+\dfrac{3}{2}x-4500$
  4. $6x-\dfrac{5}{2}x+4500$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\Rightarrow$  Here, price per unit $(p)=Rs.6$

$\Rightarrow$  Total revenue $R(x)=p.x=6x$  where $x$ is the number of unit sold.
$\Rightarrow$  Cost function $C(x)=4500+\dfrac{25}{100}R(x)$

$\Rightarrow$  Cost function $C(x)=4500+\dfrac{25}{100}\times 6x$

$\Rightarrow$   $C(x)=4500+\dfrac{3}{2}x$

$\Rightarrow$   Profit function $P(x)=R(x)-C(x)$

$\Rightarrow$  $P(x)=6x-(4500+\dfrac{3}{2}x)$

$\therefore$    $P(x)=6x-\dfrac{3}{2}x-4500$

Multiple choice business mathematics and statistics applications of calculus revenue functions from marginal revenue functions minimization of cost function and maximization of revenue function and profit function integral calculus – ii

Find the elasticity of supply when price $5$ units. Supply function is given by $q = 25 - 4p +p^2$

  1. $1$
  2. $2$
  3. $0$

  4. Cannot be determined
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$q=25-4p+p^{2}$
Elasticity of supply $=\cfrac{\cfrac{dq}{q}}{\cfrac{dp}{p}}=\cfrac{(-4+2p)\times p}{(25-4p+p^{2})}$
When $p=5$ units
Elasticity of supply $=\cfrac{(-4+10)\times 5}{(25-20+25)}=1$
Multiple choice business mathematics and statistics applications of calculus revenue functions from marginal revenue functions minimization of cost function and maximization of revenue function and profit function integral calculus – ii

Find the elasticity of supply for supply function $x = 2p^2+5$, when $p=3$.

  1. $\dfrac{23}{36}$
  2. $\dfrac{36}{23}$
  3. $\dfrac{63}{32}$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Elasticity of supply$\Rightarrow\cfrac{\cfrac{+dx}{x}}{\cfrac{dp}{p}}$$\Rightarrow\cfrac{(4p)p}{2p^{2}+5}$
Elasticity of supply when $p=3\Rightarrow\cfrac{36}{23}$
Multiple choice business mathematics and statistics applications of calculus revenue functions from marginal revenue functions minimization of cost function and maximization of revenue function and profit function integral calculus – ii

A village road is to be constructed by a team of $250$ workers. After $12$ days it was found that only $2/7^{th}$ part of the work was complete. To complete the rest in another $25$ days, how many more workers should be employed?

  1. $53$
  2. $52$
  3. $55$
  4. $50$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let the total unit of work be $7$ units.
According to question,
$\dfrac {M _{1}D _{1}}{W _{1}} = \dfrac {M _{2}D _{2}}{W _{2}} \Rightarrow \dfrac {250\times 12}{2} = \dfrac {M _{2}\times 25}{(7 - 2)} \Rightarrow M _{2} = 300$
Number of extra workers needed $= 300 - 250 = 50$.

Multiple choice business mathematics and statistics applications of calculus revenue functions from marginal revenue functions minimization of cost function and maximization of revenue function and profit function integral calculus – ii

A railway half ticket costs half the full fare but reservation charge is same. One reserved ticket from Ranchi to Howrah cost $Rs.1720$ and one full and one half ticket (both reserved) costs $Rs.2610$. Thus, the reservation charge is

  1. $Rs.60$
  2. $Rs.30$
  3. $Rs.40$

  4. None
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the full ticket cost $ Rs  \ x $and reservation charge be $ y $
As per given statements,
$ x + y = 1720 $    -(1)
And $ x + y + \frac {x}{2} + y = 2610 => \frac {3x}{2} + 2y = 2610 $ --- (2)

Multiplying equation  $ (1) $ with $ 2 $ we get, $ 2x + 2y
= 3440 $ ----- equation $ (3) $

Subtracting equation $ (2) $

from $ (3) $, we get $ \frac{x}{2} = 830 => x = 1660 $

Substituting $ x = 1660 $ in the equation $ (1) $, we get $ 1660 +y = 1720 => y = 60 $ 

Thus reservation charges were Rs $60 $

Multiple choice 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

Marginal cost is defined as the instantaneous rate of change of total cost at any level of output.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Marginal cost is mathematically defined as the derivative of the total cost function with respect to output, representing the instantaneous rate of change.

Multiple choice 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

A man spends $Rs\ 1800$ per month on an average for the first four months and $Rs\ 2000$ per month for the next $8$ months and saves $Rs\ 5600$ a year. What is his average monthly income?

  1. $Rs\ 2400$
  2. $Rs\ 2000$
  3. $Rs\ 1800$
  4. $Rs\ 2500$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Total exp in 4 months $ = 4 \times 1800 = 7200$

Total exp in 8 months $ = 8 \times 2000 = 16000$
total exp $ = 7200 + 16000$
$=23200$
savings = 5600
total income = 23200 + 5600
=28800
$Avearage\,\,income = \dfrac{{28800}}{{12}}$
$=2400/month.$