1. At 9am on November 9th, Jim is 150 miles north of Omaha and travelling south at a rate of 20 miles/hr. At the same time, Marcus is 70 miles east of Omaha and travelling east at a rate of 10 miles/hr. Finally, at the same time, Janet is 50 miles west of Omaha and travelling east at a rate of 15 miles/hr.

(a) How far apart are Jim and Marcus at 9am on November 9th?

(b) Is the distance between Jim and Marcus increasing or decreasing at 9am on November 9th? How quickly?

(c) Is the area of the triangle between Jim, Marcus, and Omaha increasing or decreasing at 9am on November 9th? How quickly?

(d) Is the distance between Jim and Marcus changing more quickly or more slowly than the distance between Marcus and Janet at 9am on November 9th?

2. A 10 meter ladder resting against a wall at a point 8 meters above the ground begins to slide down the wall at a rate of 2 meters per minute. How fast is the bottom of the ladder sliding away from the wall at that time?

3. A potter has a fixed volume of clay in the form of a cylinder. As he rolls the clay, the length of the cylinder, L, increases, while the radius, r, decreases. If the length of the cylinder is increasing at a constant rate of 0.2 cm per second, find the rate at which the radius is changing when the radius is 1.5 cm and the length is 4 cm. Recall that the volume of a cylinder of radius r and height h is given by V = ?r2h.

References:

Example. A stone thrown into a pond produces a circular ripple that expands from the point of impact. If the radius of the ripple increases at a rate of 1.5 ft/sec, how fast is the area growing when the radius is 8 feet? Know : dr = 1.5 f+ / sec Find : dA Olt r = 8 ft Equation: ( relating r + A ) A = ter 2 Take of of both sides : dA = It . 2r dr t 8 Plug in + sowe : aA = +t. 2 / 8 ) ( 1.5 ) = 1etc. 3 = 24 TC A at = 24 it f+ 2 / sec General Process for Related Rates Problems: I Draw diagram ( label variables ) and write down what you know + what you must find in terms of variables . (2 ) write down equation relating variables. 3 Take It a of both sides . I ") Plug in + solver. 5) Be sure to answer any questions asked in the problem.Note: volume equation for a cylinder would be given to you ( but areal / other triangle equations would not ) Example. A water heater that has the shape of a right cylindrical tank with a radius of 1 ft and a height of 4 ft is being drained. How fast is water draining out of the tank if the water level is dropping at 0.5 ft/min? dh Glooking for change in volume - = - 0.5 #/ min AV = ? 4 +4 r= IF this is NOT changing (r is a constant here, so we can plug it in ) V = torzh at V = TC ( 1) 2. h f+ 3 du = It - d.5 ) = 2 V = Th min at Example. A baseball player runs from home plate to first base at a rate of 15 ft/sec while another baseball player runs from third base to home plate at a rate of 17 ft/sec. At what rate is the distance between the players changing when the first baseball player is 30 ft from first base and the second baseball player is 22ft from home plate? (A baseball diamond is a 90 ft by 90 ft square.) Is the distance between the players increasing or decreasing at this time? Know : da = 15 f4 /rec db = - 17 +t / sec at 30 40 a= 10-40 = 50 #+ b = 22 #t Find: OLD at H a2+ 62 = D2 ( Pythagorean Theorem) 2 a . da alt + + 26. db Of = 2D. OP olt Need to find D 2 ( 60) ( 15 ) + 2( 22 ) (-17) = 2D. dD first ! 60 2 + 222 = D2 - D = 14084 = 43. 906 dD 2 ( led) (15 ) + 21223(-17)= 2(14084 ) at 1800- 748 = 2140842 do 524 7052 dD 524 254084 8.23 fisel