Question

a tank contains 50 liters of oil at time ( t = 4 ) hours. oil is being pumped into the tank at a rate ( r(t) ) where ( r(t) ) is measured in liters per hour, and ( t ) is measured in hours. selected values of ( r(t) ) are given in the table above. using a right riemann sum with three subintervals and data from the table, what is the approximation of the number of liters of oil that are in the tank at time ( t = 15 ) hours?

( t ) (hours)	4	7	12	15

Explanation:

Step1: Identify subintervals

The subintervals are $[4,7]$, $[7,12]$, $[12,15]$.
Calculate widths:
$\Delta t_1 = 7-4=3$, $\Delta t_2=12-7=5$, $\Delta t_3=15-12=3$

Step2: Right Riemann sum for added oil

Use right endpoints' $R(t)$ values:
$\text{Added Oil} = R(7)\Delta t_1 + R(12)\Delta t_2 + R(15)\Delta t_3$
$= 6.2\times3 + 5.9\times5 + 5.6\times3$

Step3: Compute added oil amount

Calculate each term:
$6.2\times3=18.6$, $5.9\times5=29.5$, $5.6\times3=16.8$
Sum: $18.6+29.5+16.8=64.9$

Step4: Total oil at t=15

Add initial oil to added oil:
$\text{Total Oil} = 50 + 64.9$

Answer:

114.9 liters