Question

To help restore a beach sand is being added to the beach at a rate of s(t) = 65+24sin(.3t) tons per hour, where t is measured in hourse since 5:00
a.m. how many tons of sand are added to the beach over the 3-hour period from 7:00
a.m. to 10:00
a.m.?

Answer 1

65.38 tons are added.

Using the equation, and substituting 3 for 5, we have
s(3) = 65+24sin(0.3*3) = 65.38

Answer 2

s(t) = 65+24sin(0.3t)

where t is measured in hours since 5:00 a.m

3-hour period from 7:00  a.m. to 10:00 am

5 am to 7 am = 2 hours

5 am to 10 am = 5 hours

To find the number of tons we take integral of 2 to 5  of s(t)

∫ 2  to 5 s(t) = ∫ 2  to 5 (65+24sin(0.3t))

∫65+24sin(0.3t) dt = 65t - 80 cos(0.3t) + c

Now we plug in the bounds 2 and 5

∫ 2  to 5 (65+24sin(0.3t))  = 255.36787 = 255.268

255.268 tons of sand are added to the beach over the 3-hour period