The graph of s = f(t) represents the position of an object moving along a line at time t ≥ 0. a. Assume the velocity of the object is 0 when t = 0. For what other values of t is the velocity of the object zero? b. When is the object moving in the positive direction and when is it moving in the negative direction? c. Sketch a graph of the velocity function. d. On what intervals is the speed increasing? a. The velocity of the object is zero at t = . (Use a comma to separate answers as needed.)
A water tank in the shape of an inverted circular cone has a base radius of 6 m and height of 7 m. If water is being pumped into the tank at a rate of 2 m3/min, find the rate at which the water level is rising when the water is 1.4 m deep. (Round your answer to three decimal places if required) m/min
Compute the moment of inertia of the shade region about the y-axis by integration and check answer by composite area method.
Find the volume of the region in the first octant bounded by the coordinate planes, the plane y + z = 4, and the cylinder x = 16 − y2. The volume is
Use the washer method to find the volume of the solid generated by revolving the shaded region about the x-axis.
A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure below. Use the 2nd derivative test to optimize the volume of the box.
D. If R is the base of a solid and cross sections through the solid taken perpendicular to the y-axis are squares, set up, but do not evaluate, an integral or sum of integrals that would give the volume of the solid. E. If the region R is rotated around the line y = −3, set up, but do not evaluate, an integral or sum of integrals in terms of y that would give the volume of the solid. F. If the region R is rotated around the line y = −3, set up, but do not evaluate, an integral or sum of integrals in terms of x that would give the volume of the solid.
Consider the following. the volume of the solid that lies within both the cylinder x2 + y2 = 16 and the sphere x2 + y2 + z2 = 81 Using cylindrical coordinates, write an integral that can be evaluated to find the volume V of the given solid. (Choose 0 < A ≤ 2π. Choose 0 < B.) V = ∫0 A ∫0 B ∫ −81 − r2 81 − r2 ( )dz dr dθ A = B = Find the volume.
A cylindrical water tank 6 meters high with a radius of 2 meters is buried so that the top of the tank is 1 meter below ground level (see figure). How much work is done in pumping a full tank of water up to ground level? (The water weighs 9800 newtons per cubic meter.) newton-meters
A jug of juice is taken out of the refrigerator at 11:00 and is left on the kitchen counter. The temperature of the refrigerator is 5∘C and that of the kitchen is 22∘C. One hour later, the jug of juice reaches a temperature of 11∘C. Let T(t) represent the temperature of the jug t hours after 11:00. (i) Supposing that the temperature of the jug follows Newton's Law of Cooling, determine the exact formula for T(t), by finding and solving the corresponding differential equation. T(t) = Q2 (ii) What is the temperature of the jug of juice at 13:00? Give the answer with a precision of one decimal place. Answer: Number ∘C