University:
Capital University
Course:
MATH 111 | College Algebra
Academic year:

2023
Views:

265

Pages:

2
Author:

Charlee Buchanan

Steps for Solving Related Rates Problems A related rates problem involves finding the unknown rate of change of one quantity by relating it to the already known rates of change of one or more other quantities. The following example involves relating rates of change that occur with respect to time. It is possible to relate rates of change that occur with respect to a quantity other than time. Example: “A man starts walking north at 4 ft/s from a point P. Five minutes later a woman starts walking south at 5 ft/s from a point 500 ft due east of P. At what rate are the [two] moving apart 15 minutes after the woman starts walking?” (Stewart 203) 1. Assign symbols to all quantities that are functions of time. If possible draw a diagram that describes the situation, and that includes symbols and given constant values (i.e. 500 ft). xm = distance walked north by the man Figure 2 (at t = 20 min = 1200 sec) xw = distance walked south by the woman h = distance from the man to the woman Figure 1 (at t = 5 min = 300 sec) xm h P h xm xw P 500 ft 500 ft 2. Express the given rate(s) AND the unknown rate in terms of derivatives, with respect to time, t. 3. Find an equation that relates the quantities of the given rate(s) and the unknown rate. If possible, use the geometry of the situation and relationships between quantities to eliminate variables by substitution. (xm + xw)2 + 5002 = h2 xm2 + 2xmxw + xw2 – h2 = -5002 4. Differentiate both sides of the equation with respect to time, t. 2xm + 2xm + 2xw – 2h = 0 + 2xw 5. Place the given information into the equation and solve for the unknown rate. = xm = (1200 s) = 4800 ft xw = (900 s) = 4500 ft ≈ 9313 ft h= ≈ 8.99 Practice Problems 1. “A particle is moving along the curve y = x1/2. As the particle passes through the point (4, 2), its xcoordinate increases at a rate of 3 cm/s. How fast is the distance from the particle to the origin changing at this instant?” (Stewart 203) 2. “Gravel is being dumped from a conveyor belt at a rate of 30 ft3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?” (Stewart 203) Solutions: 1. 3.02 cm/s 2. 6/(5π) ≈ 0.38 ft/min

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