Question

The rate at which rain accumulates in a bucket is modeled by the function r given by r(t)=10t−t^2, where r(t) is measured in milliliters per minute and t is measured in minutes since the rain began falling. How many milliliters of rain accumulate in the bucket from time t=0 to time t=3

Answer 1

36 milliliters of rain.

Explanation:

The rate at which rain accumluated in a bucket is given by the function:

`r(t)=10t-t^2`

Where r(t) is measured in milliliters per minute.

We want to find the total accumulation of rain from t = 0 to t = 3.

We can use the Net Change Theorem. So, we will integrate function r from t = 0 to t = 3:

`\displaystyle \int_0^3r(t)\, dt`

Substitute:

`=\displaystyle \int_0^3 10t-t^2\, dt`

Integrate:

`\displaystyle =5t^2-(1)/(3)t^3\Big|_0^3`

Evaluate:

`\displaystyle =(5(3)^2-(1)/(3)(3)^3)-(5(0)^2-(1)/(3)(0)^3)=36\text{ milliliters}`

36 milliliters of rain accumulated in the bucket from time t = 0 to t = 3.