Question

A robotic machine fills containers with an acid solution at the rate of 50 + 5 t milliliters (mL) per second, where t is in seconds and 0 ≤ t ≤ 60 . How many mL are put into a container in 60 seconds? Evaluate your answer to a whole number.

Answer 1

12,000

Explanation:

The machine fills the containers at a rate of 50+5t milliliters (mL) per second.

Therefore, the rate of change of the number of containers, N is:

`(dN)/(dt)=50+5t, 0\leq t\leq 60`

`dN=(50+5t)dt\\$Taking integrals of both sides\\\int dN=\int (50+5t)dt\\N(t)=50t+(5t^2)/(2)+C $(C a constant of integration)\\\\When t=0, , No containers are filled, therefore:$ N(t)=0\\0=50(0)+(5(0)^2)/(2)+C\\C=0\\$Therefore, N(t)=50t+2.5t^2`

When t=60 seconds

`N(60)=50(60)+2.5(60)^2\\N(60)=12000$ mL`

Therefore, 12,000 milliliters of acid solution are put into a container in 60 seconds.