Did the workers get their bonus?

Question

After 2 hours Vanessa realized her company had only made 2 units of product. She wanted to encourage more work, so she told them if they can make a total of 20 by the end of the day (in 6 hours) she would give them all a bonus. As they worked, it turned out the rate at which they were producing units could be modeled by the function r(t)=0.1t^3-1.1t^2+2.4t+3.6, 0<_t<_6 Where t=0 is the time she started the challenge and t=6 is the end of the workday.

Did the workers get their bonus?

<_=greater than or equal to

Answer 1

To determine the number of units produced during the 6 hours between the beginning of the challenge and the end of the day, we can integrate the rate at which the units are being produced:

∫₀⁶(0.1t³ - 1.1t² + 2.4t + 3.6)dt = [(1/4)0.1t⁴ - (1/3)1.1t³ + (1/2)2.4t² + 3.6t]₀⁶

= (1/4)0.1(6)⁴ - (1/3)1.1(6)³ + (1/2)(2.4)(6)² + 3.6(6)

= (1/4)(0.1)(6)⁴ - (1/3)(1.1)(6)³ + (1/2)(0.4)(6)³ + 0.1(6³)

= 6³((0.1)6/4 - (1/3)(1.1) + (1/2)(0.4) + 0.1)

= 6³(0.15 - (1/3)(1.1) + 0.2 + 0.1)

= 6³(0.45 - 1.1/3)

= 6³(1.35 - 1.1)/3

= 2(6²(0.25)

= 2(3²)

= 18

(no need to follow these exact steps, as long as you can simplify your answer and wind up with 18).

18 was the exact amount needed in this period to obtain the bonus (there had to be 20 total and 2 were already produced).

Therefore, they should get their bonus