Question

Alabama Instruments Company has set up a production line to manufacture a new calculator. The rate of production of these calculators after t weeks is given by the formula below. (Notice that production approaches 5000 per week as time goes on, but the initial production is lower because of the workers' unfamiliarity with the new techniques.) dx/dt = 5000 (1 - 100/(t + 12)2) Find the number of calculators produced from the beginning of the third week to the end of the fourth week. (Round your answer to the nearest integer.) calculators

Answer 1

Answer:

5536 calculators

Step-by-step explanation:

We integrate the function dx/dt to obtain the number of new calculators between beginning of the 3rd week  and end of week 4. Note that beginning of 3rd week is the same as end of 2nd week. So, $\int\limits^4_2 {\frac{dx}{dt} } \, dx$=  $\int\limits^4_2 {5000 (1 - \frac{100}{(t +12)^{2} }) } \, dx$

Let u = t + 12, then $\frac{du}{dt}$= 1. So, du = dt. We also change the limits of our integration. So, when t = 2, u = 2 + 12 = 14 and when t = 4, u = 4 + 12 = 16

Then  $\int\limits^4_2 {5000(1 -\frac{100}{(t+12)^{2} } )} \, dx$ = ∫₁₄¹⁶ ${5000(1 -\frac{100}{u^{2} } )} \, du = 5000[ u + \frac{100}{u} ]$₁₄¹⁶ = $5000[16 + \frac{100}{16} - (14 + \frac{100}{14} )] = 5000 [16 - 14 + \frac{100}{16} - \frac{100}{14} ] = 5000 [2 + \frac{100}{16} - \frac{100}{14} ] = 5535.7$

≈ 5536 calculators