Verified step by step guidance
04:48
04:18Emptying a conical tank A water tank is shaped like an inverted cone with height 6 m and base radius 1.5 m (see figure).
a. If the tank is full, how much work is required to pump the water to the level of the top of the tank and out of the tank?
17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.
a. Determine the position function, for t≥0, using the antiderivative method
v(t) = −t³+3t²−2t on [0, 3]; s(0)=4
Critical depth A large tank has a plastic window on one wall that is designed to withstand a force of 90,000 N. The square window is 2 m on a side, and its lower edge is 1 m from the bottom of the tank.
a. If the tank is filled to a depth of 4 m, will the window withstand the resulting force?
Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N'(t) = r+A sin 2πt/P, where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t=0.
a. Suppose P=10, A=20, and r=0. If the initial population is N(0)=10, does the population ever become extinct? Explain.
Mass of two bars Two bars of length L have densities ρ₁(x) = 4e^−x and ρ₂(x) = 6e^−2x, for 0≤x≤L.
a. For what values of L is bar 1 heavier than bar 2?
Determine whether the following statements are true and give an explanation or counterexample.
a. If the curve y=f(x) on the interval [a, b] is revolved about the y-axis, the area of the surface generated is ∫f(b)f(a) 2πf(y)√1+f′(y)^2 dy.
