Emanuele

A ball of radius 15 has a round hole of radius 9 drilled through its center. Find the volume of the resulting solid.

The radius of a right circular cone is increasing at a rate of 4 inches per second and its height is decreasing at a rate of 5 inches per second. At what rate is the volume of the cone changing when the radius is 30 inches and the height is 30 inches?

In a simple electric circuit, Ohm's law states that V=IR, where V is the voltage in volts, I is the current in amperes, and R is the resistance in ohms. Assume that, as the battery wears out, the voltage decreases at 0.02 volts per second and, as the resistor heats up, the resistance is increasing at 0.01 ohms per second. When the resistance is 200 ohms and the current is 0.01 amperes, at what rate is the current changing?

A particle is moving along the curve y=5sqrt(3x+3). As the particle passes through the point (2,15), its x-coordinate increases at a rate of 4 units per second. Find the rate of change of the distance from the particle to the origin at this instant.

At noon, ship A is 20 nautical miles due west of ship B. Ship A is sailing west at 20 knots and ship B is sailing north at 25 knots. How fast (in knots) is the distance between the ships changing at 4 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)

A plane flying with a constant speed of 4 km/min passes over a ground radar station at an altitude of 5 km and climbs at an angle of 35 degrees. At what rate, in km/min is the distance from the plane to the radar station increasing 5 minutes later?

A spherical snowball is melting in such a way that its diameter is decreasing at rate of 0.1 cm/min. At what rate is the volume of the snowball decreasing when the diameter is 11 cm. (Note the answer is a positive number).

A street light is at the top of a 10 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 8 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 40 ft from the base of the pole?

Note: You should draw a picture of a right triangle with the vertical side representing the pole, and the other end of the hypotenuse representing the tip of the woman's shadow. Where does the woman fit into this picture? Label her position as a variable, and label the tip of her shadow as another variable. You might like to use similar triangles to find a relationship between these two variables.

Helium is pumped into a spherical balloon at a rate of 4 cubic feet per second. How fast is the radius increasing after 3 minutes?

Note: The volume of a sphere is given by V=(4/3)r^3.

Rate of change of radius (in feet per second) = ?

A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at a rate of 2 feet per second, how fast is the circumference changing when the radius is 17 feet?

Change in circumference = ?

At noon, ship A is 20 nautical miles due west of ship B. Ship A is sailing west at 25 knots and ship B is sailing north at 19 knots. How fast (in knots) is the distance between the ships changing at 6 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)

Note: Draw yourself a diagram which shows where the ships are at noon and where they are "some time" later on. You will need to use geometry to work out a formula which tells you how far apart the ships are at time t, and you will need to use "distance = velocity * time" to work out how far the ships have travelled after time t.

xy^3+xy=4 at the point (2,1). The equation of this tangent line can be written in the form y=mx+b where m=? and b=?

A point (x,y) in the plane moves in such a way that twice its distance from the origin plus its distance from the point (9,0) is constant. Find the equation of the tangent to the locus of the point as it passes through (4,4). The equation of this tangent line can be written in the form y=mx+b

m = ?

b = ?

*please show your work :)

A lighthouse is located on an island 5 km. away from the nearest point P on a straight shoreline and its light makes 4 revolutions per minute. The point where the beam meets the shoreline is moving at ______ km./sec. when this point is 2 km. from the point P.