In Cartesian coordinates, the region is given by

,

, and

. Converting to cylindrical coordinates, using

we get a Jacobian determinant of

, and the region is given in cylindrical coordinates by

,

, and

.
The volume is then
Answer:
40: 1, 2, 4, 5, 8, 10, 20, 40
150: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150
Step-by-step explanation:
Answer:
C. 2.0 < t < 2.5
Step-by-step explanation:
time = distance / speed
The circumference of the lake is given by ...
C = πd = 2π miles ≈ 6.28 miles
Then Johanna's time is ...
(6.28 mi)/(3 mi/h) ≈ 2.09 h
This time is in the interval (2, 2.5), so matches choice C.
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<em>Alternate solution</em>
If we take pi to be 3, then this boils down to ...
2×3/3 = 2 . . . hours
Pi is on the order of 5% more than 3, so her time will be on the order of 5% more than 2 hours, or just above 2, but not as great as 2.5 hours. This sort of estimating can get you to the correct answer without a calculator.
Use the the double angle formula:
sin(2A)=2sin(A)cos(A)
substitute 2x for A, then
20sin(2x)cos(2x)=10(sin(2(2x))cos(2(2x))=10sin(4x)