Andrew K. answered • 01/09/15

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Hi Jennifer,

I always like to start related rates problems by drawing a picture, identifying variables, and assigning numerical values to what is known. I'm going to ignore the statement about the person being 5 ft tall, because the problem doesn't ask specifically about the distance between the UFO and your feet, or the UFO and your head - which is really the only thing that would make that 5 ft have any significance. Instead, I'll just use the information that the UFO starts 500 ft away from you, horizontally, and 300 ft above the ground.

**Step 1: Draw a picture, identify variables, and assign numerical values to what is known.**

UFO

/ |

/ |

/ |

R / | y

/ |

/ |

You /_______|

**x**

**where "x" and "y" are the horizontal and vertical distances between you and the UFO, and R is the straight line distance.**

Initially, x=500 ft and y=300 ft

Since you know that the UFO is moving toward you at 30 ft/s, we could say that dx/dt = -30 ft/s (x gets smaller as the UFO gets closer.

**Step 2: Develop an equation that relates the variables you are interested in.**

The three sides of a right triangle can always be related using the Pythagorean Theorem

R

^{2}= x^{2}+ y^{2}**Step 3: Determine the values of variables for the specific moment you are being asked about.**

We are specifically being asked about the rate of change of R (dR/dt) after 10 seconds. After those 10 seconds, the "y" value would still be 300 ft, because the altitude is not changing, but the "x" value would now be:

500 ft + (-30 ft/s)*10s = 200 ft.

So, using our equation from step 2, the straight line distance would be:

R

^{2}= x^{2}+ y^{2} R = √(x

^{2}+ y^{2}) R = √(200

^{2}+ 300^{2}) R = 361 ft (rounded to 3 digits)

**Step 4: Use implicit differentiation on the equation developed in Step 2:**

**R**

^{2}= x

^{2}+ y

^{2}

2R*(dR/dt) = 2x*(dx/dt) + 2y*(dy/dt)

Since we know that the "y" value does not change, dy/dt is 0.

2R*(dR/dt) = 2x*(dx/dt)

**Step 5: Solve for the desired variable, and plug in the known values:**

**dR/dt = (x*(dx/dt))/R**

dR/dt = (200 ft * (-30 ft/s))/(361 ft)

dR/dt = -16.6 ft/s

So, at that specific moment, the straight line distance between you and the UFO is decreasing (because the answer is negative) at a rate of 16.6 ft/s.

I hope this helps!

Andy

Andrew K.

tutor

You're welcome, please let me know if you have any questions!

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01/09/15

Jennifer W.

01/09/15