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06:46Variable 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.
b. Use the Chain Rule to show that dv/dt = 1/2 d/dy(v²).
Determine whether the following statements are true and give an explanation or counterexample.
b. The area of the region between y=sin x and y=cos x on the interval [0,π/2] is ∫π/20(cosx−sinx)dx.
Two runners At noon (t=0), Alicia starts running along a long straight road at 4 mi/hr. Her velocity decreases according to the function v(t) = 4 / t + 1 for t≥0. At noon, Boris also starts running along the same road with a 2-mi head start on Alicia; his velocity is given by u(t) = 2 / t + 1, for t≥0. Assume t is measured in hours.
b. When, if ever, does Alicia overtake Boris?
Filling a tank A 2000-liter cistern is empty when water begins flowing into it (at t=0 at a rate (in L/min) given by Q′(t) = 3√t, where t is measured in minutes.
b. Find the function that gives the amount of water in the tank at any time t≥0.
A torus (doughnut) A torus is formed when a circle of radius 2 centered at (3, 0) is revolved about the y-axis.
b. Use the washer method to write an integral for the volume of the torus.
Cycling distance A cyclist rides down a long straight road with a velocity (in m/min) given by v(t) = 400−20t, for 0≤t≤10, where t is measured in minutes.
b. How far does the cyclist travel in the first 10 min?
