Question

the rate at which the amount of water in a tank is changing, in liters per hour, can be modeled by the function ( r(t) ), where ( t ) is measured in hours. selected values of ( r(t) ) are shown in the table below. approximate the average value of ( r(t) ) over the interval ( 0, 16 ) using a left riemann sum with 4 subintervals indicated by the table. you may use a calculator if necessary.

( t )	0	5	8	11	16

Explanation:

Step1: Determine subinterval widths

The interval is \([0, 16]\) with 4 subintervals. The widths are calculated as \( \Delta t_1 = 5 - 0 = 5 \), \( \Delta t_2 = 8 - 5 = 3 \), \( \Delta t_3 = 11 - 8 = 3 \), \( \Delta t_4 = 16 - 11 = 5 \).

Step2: Identify left endpoints and their R(t) values

Left endpoints are \( t = 0, 5, 8, 11 \) with \( R(t) = 10, 9, 3, 2 \) respectively.

Step3: Calculate left Riemann sum

Sum \( = R(0)\Delta t_1 + R(5)\Delta t_2 + R(8)\Delta t_3 + R(11)\Delta t_4 \)
\( = 10\times5 + 9\times3 + 3\times3 + 2\times5 \)
\( = 50 + 27 + 9 + 10 = 96 \)

Step4: Find average value

Average value \( = \frac{\text{Total sum}}{\text{Interval length}} = \frac{96}{16} = 6 \)

Answer:

6