Sand delivered to a construction site

Name: ______________________ Date: ____________ Sand delivered to a construction site (9 points)

Table

Sand is delivered to a construction site at a rate

S(t)

tons per hour, where

t

is measured in hours and

0 \le t \le 8

S

is differentiable. Selected values of

S(t)

are shown in the table. For parts (d) and (e), the rate is modeled by

M(t) = 4 + 3.2\sin\left(\frac{t}{3}\right)

tons per hour.

Rate S(t), in tons per hour, at selected times t, in hours
$t$ (hours)	0	1	4	6	8
$S(t)$ (tons per hour)	4.0	5.1	7.1	6.9	5.4

(a) Use a right Riemann sum with the four subintervals indicated by the table to approximate $\int_0^8 S(t)\,dt$ . Show the computations that lead to your answer, and indicate units of measure. [2 points]
(b) Explain the meaning of $\int_0^8 S(t)\,dt$ in the context of the problem. [1 point]
(c) Must there be a time $t$ , $1 < t < 4$ , at which $S(t) = 6$ ? Justify your answer. [2 points]
(d) Using the model $M$ , find the total amount of sand delivered from $t = 0$ to $t = 8$ hours. [2 points]
(e) Using the model $M$ , find the time $T$ at which the total amount of sand delivered since $t = 0$ reaches 25 tons. [2 points]

The question