Chapter 1.1, Problem 38E

A ladder 27 feet long leans against a wall and the foot of the ladder is sliding away at a constant rate of 3 feet/sec. Meanwhile, a firefighter is climbing up the ladder at a rate of 2 feet/sec. When the firefighter has climbed up 6 feet of the ladder, the ladder makes an angle of л/3 with the ground. Answer the two related rates questions below. (Hint: Use two carefully labeled similar right triangles.) (a) If h is the height of the firefighter above the ground, at the instant the angle of the ladder with the ground is л/3, find dh/dt= feet/sec. (b) If w is the horizontal distance from the firefighter to the wall, at the instant the angle of the ladder with the ground is л/3, find dw/dt= feet/sec.

Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate (in mi/h) is the distance between the cars increasing four hours later? Step 1 Using the diagram of a right triangle given below, the relation between x, y, and z is z² = x²+ +12 x Step 2 We must find dz/dt. Differentiating both sides and simplifying gives us the following. 2z dz dt dx 2x. +2y dt dx dy dz x +y dt dt dt 2z dy dt × dx (x+y dt dy dt

An elastic rope is attached to the ground at the positions shown in the picture. The rope is being pulled up along the dotted line. Assume the units are meters. 9 ground level Assume that x is increasing at a rate of 3 meters/sec. (a) Write as a function of x: 0= (b) When x=10, the angle is changing at a rate of rad/sec. (c) Let L be the the left hand piece of rope and R the right hand piece of rope. When x=10, is the rate of change of L larger than the rate of change of R? ○ Yes ○ No