Differentiation Rules 250 CHAPTER 3 the shape of a cone always equal. How fas when the pile is 10 ft hig 21. The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of 2 cm2/min. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 cm2? 22. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s, how fast is the boat approaching the dock when it is 8 m from the dock? 30. A kite 100 ft above the ground moves horizontally of 8 ft/s. At what rate is the angle between the string horizontal decreasing when 200 ft of string has been le oon, ship A is 100 km west of ship B. Ship A is sailing at 35 km/h and ship B is sailing north at 25 km/h. is the distance between the ships changing at 31. The sides of an equilateral triangle are increasing at a rate 10 cm/min. At what rate is the area of the triangle increasin when the sides are 30 cm long? ves along the curve y 2 sin(Tx/2). As the hrough the point (1), its x-coordinate of /10 cm/s. How fast is the distance he origin changing at this instant? 32. How fast is the angle between the ladder and the ground changing in Example 2 when the bottom of the ladder is 6 ft an inverted conical tank at a rate of ne time that water is being pumped te. The tank has height 6 m and If the water level is rising at a ht of the water is 2 m, find from the wall? 33. The top of a ladder slides down a vertical wall at a rate of 0.15 m/s. At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s. How long is the ladder? nped into the tank. 34. According to the model we used to solve Example 2, what happens as the top of the ladder approaches the ground? Is the model appropriate for small values of y? e the shape of isos- op and have a height ter at a rate of when the water 35. If the minute hand of a clock has length r (in centimeters), find the rate at which it sweeps out area as a function of r. s the 36. A faucet is filling a hemispherical basin of diameter 60 cm with water at a rate of 2 L/min. Find the rate at which the water is rising in the basin when it is half full. [Use the following facts: 1 L is 1000 cm. The volume of the portion of a sphere with radius r from the bottom to a height h is T (rh2 -3h), as we will show in Chapter 6.] he the V = 37. Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure P and volume V satisfy the equation PV= C, where C is a constant. Sup that at a certain instant the volume is 600 cm, the is 150 kPa, and the pressure is increasing at a rat kPa/min. At what rate is the volume decr nt? r expands adiabatically (K ressure P and volury y14 = C, whe t the v