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Time Rates Problem

# Time Rates Problem - Application 4 RELATED RATES TIME RATES...

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Application 4 RELATED RATES/ TIME RATES PROBLEMS

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OBJECTIVES: At the end of the lesson, the students should be able to: 1. Apply the concept of derivatives as a rate of change. 2. Solve different types of rate of change problems O
If a particle is moving along a straight line according to the equation of motion , since the velocity may be interpreted as a rate of change of distance with respect to time, thus we have shown that the velocity of the particle at time “t” is the derivative of “s” with respect to “t”. ) t ( f s = There are many problems in which we are concerned with the rate of change of two or more related variables with respect to time, in which it is not necessary to express each of these variables directly as function of time. For example, we are given an equation involving the variables x and y, and that both x and y are functions of the third variable t, where t denotes time.

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Since the rate of change of x and y with respect to t is given by and , respectively, we differentiate both sides of the given equation with respect to t by applying the chain rule . When two or more variables, all functions of t, are related by an equation, the relation between their rates of change may be obtained by differentiating the equation with respect to t. dx dt dy dt
A Strategy for Solving Related Rates Problems (p. 205)

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Example 1 A 17 ft ladder is leaning against a wall. If the bottom of the ladder is pulled along the ground away from the wall at the constant rate of 5 ft/sec, how fast will the top of the ladder be moving down the wall when it is 8 ft above the ground? ( 29 ( 29 ( 29 wall the from away ground the along pulled is ladder the of bottom the since sec time t instant any at ground the from ladder the of top the of ft distance y instant any at wall the from ladder the of bottom the of ft distance x Let = = = sec ft 5 dt dx = x 17 ft.
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