Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. If water flows into a tank at a constant rate (for example, 6 gal/min), the volume of water in the tank increases according to a linear function of time.

Displacement and distance from velocity Consider the graph shown in the figure, which gives the velocity of an object moving along a line. Assume time is measured in hours and distance is measured in miles. The areas of three regions bounded by the velocity curve and the t-axis are also given.

a. On what intervals is the object moving in the positive direction?

Variable gravity At Earth’s surface, the acceleration due to gravity is approximately g=9.8 m/s² (with local variations). However, the acceleration decreases with distance from the surface according to Newton’s law of gravitation. At a distance of y meters from Earth’s surface, the acceleration is given by a(y) = - g / (1+y/R)², where R=6.4×10⁶ m is the radius of Earth.

f. Graph ymax as a function of v0. What is the maximum height when v0=500 m/s,1500 m/s, and 5 km/s?

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. Given only the velocity of an object moving on a line, it is possible to find its displacement, but not its position.

70–72. Variable density in one dimension Find the mass of the following thin bars.

A bar on the interval 0≤x≤6 with a density ρ(x) = {1 if 0 ≤ x < 2

2 if 2 ≤ x < 4

4 if 4 ≤ x ≤ 6

Fuel consumption A small plane in flight consumes fuel at a rate (in gal/min) given by

R'(t) ={ 4t^{1/3} if 0 ≤ t ≤ 8 (take-off)

2 if t> 0 (cruising)

a. Find a function R that gives the total fuel consumed, for 0≤t≤8.

21–30. {Use of Tech} Arc length by calculator

a. Write and simplify the integral that gives the arc length of the following curves on the given interval. 

y = cos 2x, for 0 ≤ x ≤ π