A dog is walking at a rate of 5 𝑚𝑝ℎ (miles per hour) towards the bottom of a 60-foot tree

Question

Problem 2: A dog is walking at a rate of 5 𝑚𝑝ℎ (miles per hour) towards the bottom of a 60-foot tree. At what rate is the dog approaching the top of the tree when he is 80 feet away from the tree? (Ignore the dog’s height.) *For the Check/Assess portion of your solution, either: • comment on the answer you found. Does it seem reasonable? Why or why not? OR • comment on how considering the dog’s height would change the answer you found above.

Answer 1

Note that you don't even need to converts miles to feet or hours into minutes or seconds.

I left out the units in the video in the interest of time but where we have

2x dx/dt + 2y dy/dt = 2h dh/dt

we have

2(80 feet)(-5 miles/hour )+ 2(60 feet)(-5 /miles/hour) = 2(100 feet) dh/dt

and

-800 (feet x miles)/hour = 200 feet dh/dt

after which dividing both sides by 200 feet we get

-4 miles/hour =dh/dt

which make sense. The distance from the dog to the top of the tree is decreasing by 4 miles per hour.

Answer 2

the problem involves a right triangle with sides 80 and 60

distance from the dog to the top of the tree is the hypotenuse of that right triangle = sqr(80^2+60^2) = 100 feet. It's a 3-4-5 right triangle multiplied by a factor of 20

h^2 = 60^2 + b^2 where h= distance from dog to top of the tree and b = distance of the dog to the bottom of the tree.

b' = speed of the dog= 5mph, h' = speed or rate of change of the distance to the top of the tree

take the derivative

2hh' = 0 + 2bb'

hh' = bb'

h' = bb'/h

= (80)5/100 = 400/100 = 4

h' = 4 mph= rate of change or speed of how much closer the dog is getting to the top of the tree

That's an instantaneous speed at a point in time only.

as the dog gets closer, that speed decreases

if the dog were 60 feet tall, the speed of 5mph would also be equal to how fast he gets closer to the top of the tree

the average rate of speed is different

the dog gets 80 feet closer to the bottom of the tree in the same time as it gets 40 feet closer to the top of the tree (100-60= 40). that's a speed of 1/2 of 5mph = 2.5 mph for the hypotenuse of the right triangle or the distance to the top of the tree.

the dog is 80 feet away, going 5 mph.

1 mile = 5280 feet

5 mph = 5(5280) = 26,400 feet per hour

= 26,400/60 = 4,40 feet per minute

= 440/60 = 44/6 = 22/3 = 7 1/3 feet per second

the dog is 80 feet away. time to reach the tree is 80/(22/3) = 80(3/22) = 240/22 = 120/11 = 10 10/11 seconds

at that time the dog will be 60 feet from the top of the tree

meaning the dog went from 100 to 60 feet = 40 feet closer in 10 10/11 seconds. the dog's speed relative to the top of the tree is 40 feet per 10 10/11 seconds or 40 feet per 120/11 seconds = 40/(120/11) = 40(11/120) = 11/3 feet per second

11/3 is 1/2 of 22/3 feet per second

1/2 of 5 mph = 5/2 = 2.5 mph

but at the point in time when the dog is 80 feet away, the speed of the dog relative to the top of the tree is 4 mph