Question

Alabama Instruments Company has set up a production line to manufacture a new calculator. The rate of production of these calculators after\({\rm{t}}\)weeks is

\(\frac{{{\rm{dx}}}}{{{\rm{dt}}}}{\rm{ = 5000}}\left( {{\rm{1 - }}\frac{{{\rm{100}}}}{{{{{\rm{(t + 10)}}}^{\rm{2}}}}}} \right)\)

calculators/week
(Notice that production approaches \({\rm{5000}}\) per week as time goes on, but the initial production is lower because of the workers' unfamiliarity with the new techniques.) Find the number of calculators produced from the beginning of the third week to the end of the fourth week.

Short answer

Calculator created between the start of the third week and the end of the fourth week\({\rm{ = 4047}}\).

Step-by-step solution

Given Data.

\(\frac{{{\rm{dx}}}}{{{\rm{dt}}}}{\rm{ = 5000}}\left( {{\rm{1 - }}\frac{{{\rm{100}}}}{{{{{\rm{(t + 10)}}}^{\rm{2}}}}}} \right)\)

\({\rm{5000}}\)per week.

Integrating the value.

\(\begin{array}{c}\frac{{{\rm{dx}}}}{{{\rm{dt}}}}{\rm{ = 5000}}\left( {{\rm{1 - }}\frac{{{\rm{100}}}}{{{{{\rm{(t + 10)}}}^{\rm{2}}}}}} \right)\\{\rm{dx = 5000}}\left( {{\rm{1 - }}\frac{{{\rm{100}}}}{{{{{\rm{(t + 10)}}}^{\rm{2}}}}}} \right){\rm{dt}}\end{array}\)

Integrate from \({\rm{2}}\) to \({\rm{4}}\)because the start of the third week is the same as the end of the second.

\(\begin{array}{c}\int_{{\rm{x(2)}}}^{{\rm{x(4)}}} {\rm{d}} {\rm{x = }}\int_{\rm{2}}^{\rm{4}} {\rm{5}} {\rm{000}}\left( {{\rm{1 - }}\frac{{{\rm{100}}}}{{{{{\rm{(t + 10)}}}^{\rm{2}}}}}} \right){\rm{dt}}\\{\rm{x(4) - x(2) = }}\int_{\rm{2}}^{\rm{4}} {\rm{5}} {\rm{000}}\left( {{\rm{1 - }}\frac{{{\rm{100}}}}{{{{{\rm{(t + 10)}}}^{\rm{2}}}}}} \right){\rm{dt}}\end{array}\)

Substitute\({\rm{t + 10 = u}}\)and \({\rm{d t = d u}}\)

Limits of Integration will change from\(\int_{\rm{ - }} {{{\rm{2}}^{\rm{4}}}} \)to\(\int_{10 + 2}^{10 + 4} = \mathop \smallint \nolimits_{12}^{14} \)

\(\begin{array}{c}{\rm{x(4) - x(2) = }}\int_{{\rm{12}}}^{{\rm{14}}} {\rm{5}} {\rm{000}}\left( {{\rm{1 - }}\frac{{{\rm{100}}}}{{{{\rm{u}}^{\rm{2}}}}}} \right){\rm{du}}\\{\rm{x(4) - x(2) = 5000}}\left( {{\rm{u + }}\frac{{{\rm{100}}}}{{\rm{u}}}} \right)_{{\rm{12}}}^{{\rm{14}}}\\{\rm{x(4) - x(2) = 5000}}\left( {{\rm{14 + }}\frac{{{\rm{100}}}}{{{\rm{14}}}}} \right){\rm{ - 5000}}\left( {{\rm{12 + }}\frac{{{\rm{100}}}}{{{\rm{12}}}}} \right)\\ \approx {\rm{4047}}{\rm{.619}}\end{array}\)

Therefore, between the start of the third week and the end of the fourth week, a calculator was created \({\rm{ = 4047}}\).