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

Answer:

36 milliliters of rain.

Step-by-step 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:

$\displaystyle =5t^2-\frac{1}{3}t^3\Big|_0^3$

Evaluate:

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

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