I have a question about related rates and basically i need help checking my work

Question

You, five feet tall, are in your backyard stargazing when you notice a bright light in a field 500 feet away from you and 300 feet in the sky. You focus your telescope on it and notice that it is a UFO making its way across the sky towards where you are standing at a speed of 30 feet per second. What is the rate of change in the distance between you and the UFO after the UFO has flown for 10 seconds?

Answer 1

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

 

    R² = x² + y²

 

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:

 

    R² = x² + y²

 

    R = √(x² + y²)

 

    R = √(200² + 300²)

 

    R = 361 ft (rounded to 3 digits)

 

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

 

    R² = x² + y²

 

    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