Question

At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production P w.r.t. additional number of workers x is given by-
$\dfrac{{d{\text{P}}}}{{d{\text{x}}}} = 100 - 12\sqrt {\text{x}}$
If the firm employs 25 more workers, then the new level of production of items is-
A. 3000
B. 3500
C. 4500
D. 2500

Answer 1

Hint: We will use integration to solve this problem because we cannot find the change in production of items directly by subtracting the final and initial values. This is because the change in production depends on the number of workers. The formula used will be-
$\smallint {{\text{x}}^{\text{n}}}dx = \dfrac{{{{\text{x}}^{{\text{n}} + 1}}}}{{{\text{n}} + 1}} + {\text{c}}$

Complete step-by-step answer:
It is given that at 2000 items, the number of extra workers is 0. Let the final number of items be P. At P items, the number of extra workers are 25. These are the limits of our integration. The change id production can be written as-

$dP = 100 - 12\sqrt {\text{x}} dx$
$\mathop \smallint \nolimits_{2000}^{\text{P}} dP = \mathop \smallint \nolimits_0^{25} \left( {100 - 12\sqrt {\text{x}} } \right)dx$
${\text{P}} - 2000 = \left( {100x - \dfrac{{12{{\text{x}}^{\dfrac{3}{2}}}}}{3} \times 2} \right)_0^{25}$
${\text{P}} - 2000 = \left( {100 \times 25 - 8 \times 125} \right) - 0$
P - 2000 = 2500 - 1000
P = 3500

This is the number of items when 25 workers are added are 3500. The correct option is B. 3500.

Note: In such types of questions, read the question carefully about what is asked and what are the limits that have to be substituted in the integrals. Also, we don’t need to add constant of integration in definite integration.