Question

W(x) represents the weight of the sand in pounds on a scale x seconds after we started to unload the sand. r(x) represents the rate at which the weight of the sand changes with respect to time, in pounds per seconds.

Which of the folloing expressions (check ALL) give the exact net chnage in the weight of the sand, in pounds, between x= 3 seconds and x = 6 seconds?

a. w(6) - w(3)
b. w (3) + ⁶∫₃ r (x) dx
c. ⁶∫₃ r (x) dx
d. r(6) - r(3)
e. ⁶∫₃ (x) dx

Answer 1

The net change in the weight of the sand between x=3 seconds and x=6 seconds is correctly represented by the expressions W(6) - W(3) and ¶∫₃ r(x) dx.

Step-by-step explanation:

The net change in the weight of the sand between x=3 seconds and x=6 seconds can be determined using either the function representing the weight of the sand or the rate of change of the weight of the sand. The expression W(6) - W(3) gives the net change in weight by directly calculating the difference in weights at the two times. Alternatively, one can use the integral of the rate of change function, ¶∫₃ r(x) dx, which gives the net change in weight by summing up all the tiny changes in weight over the interval from 3 to 6 seconds.

Therefore, the correct expressions for calculating the exact net change in the weight of the sand are:

• W(6) - W(3)
• ¶∫₃ r(x) dx

These expressions are based on the fact that weight is mass times the acceleration due to gravity, as indicated by w = mg. Since r(x) represents the rate of weight change, the integral of r(x) over the time period will give the overall change in weight.