You need to know the speed and direction of object
Answer:
The rate of change of the distance between the helicopter and yourself (in ft/s) after 5 s is
ft/ sec
Explanation:
Given:
h(t) = 25 ft/sec
x(t) = 10 ft/ sec
h(5) = 25 ft/sec . 5 = 125 ft
x(5) = 10 ft/sec . 5 = 50 ft
Now we can calculate the distance between the person and the helicopter by using the Pythagorean theorem
![D(t) = \sqrt{h^2 + x^2}](https://tex.z-dn.net/?f=D%28t%29%20%3D%20%5Csqrt%7Bh%5E2%20%2B%20x%5E2%7D)
Lets find the derivative of distance with respect to time
![\frac{dD}{dt} (t) = \frac{2h \cdot \frac{dh}{dt} +2x \cdot\frac{dx}{dt}} {2\sqrt{h^2 + x^2}}](https://tex.z-dn.net/?f=%5Cfrac%7BdD%7D%7Bdt%7D%20%28t%29%20%20%3D%20%5Cfrac%7B2h%20%5Ccdot%20%5Cfrac%7Bdh%7D%7Bdt%7D%20%2B2x%20%5Ccdot%5Cfrac%7Bdx%7D%7Bdt%7D%7D%20%7B2%5Csqrt%7Bh%5E2%20%2B%20x%5E2%7D%7D)
Substituting the values of h(t) and x(t) and simplifying we get,
![\frac{dD}{dt}(t) = \frac{50t \cdot \frac{dh}{dt} + 20 \cdot \frac{dx}dt}{2\sqrt{625\cdot t^2 + 100 \cdot t^2}}](https://tex.z-dn.net/?f=%5Cfrac%7BdD%7D%7Bdt%7D%28t%29%20%3D%20%5Cfrac%7B50t%20%5Ccdot%20%5Cfrac%7Bdh%7D%7Bdt%7D%20%2B%2020%20%5Ccdot%20%5Cfrac%7Bdx%7Ddt%7D%7B2%5Csqrt%7B625%5Ccdot%20t%5E2%20%2B%20100%20%5Ccdot%20t%5E2%7D%7D)
![\frac{dh}{dt} = 25ft/sec](https://tex.z-dn.net/?f=%5Cfrac%7Bdh%7D%7Bdt%7D%20%3D%2025ft%2Fsec)
![\frac{dx}{dt} = 10 ft/sec](https://tex.z-dn.net/?f=%5Cfrac%7Bdx%7D%7Bdt%7D%20%3D%2010%20ft%2Fsec)
=
=
ft / sec
Answer:
Your friend is 2.143 blocks from the restaurant.
You are 2.857 blocks from the restaurant.
Explanation:
Let t be the time both you and your friend take to walk to the restaurant.
The distance (m) from your building to the restaurant is your walking time t times your speed v1
![s_1 = v_1t = 1.6t](https://tex.z-dn.net/?f=s_1%20%3D%20v_1t%20%3D%201.6t)
Similarly the distance (m) from your friend building to the restaurant:
![s_2 = v_2t = 1.2t](https://tex.z-dn.net/?f=s_2%20%3D%20v_2t%20%3D%201.2t)
Let b be the length (in m) of a block, the total distance of 5 blocks is 5b
![s_1 + s_2 = 5b](https://tex.z-dn.net/?f=s_1%20%2B%20s_2%20%3D%205b)
![1.6t + 1.2t = 5b](https://tex.z-dn.net/?f=1.6t%20%2B%201.2t%20%3D%205b)
![2.8t = 5b](https://tex.z-dn.net/?f=2.8t%20%3D%205b)
![t = 5b/2.8 = 25b/14](https://tex.z-dn.net/?f=t%20%3D%205b%2F2.8%20%3D%2025b%2F14)
![s_2 = 1.2t = 1.2(25b/14) = 2.143b](https://tex.z-dn.net/?f=s_2%20%3D%201.2t%20%3D%201.2%2825b%2F14%29%20%3D%202.143b)
So your friend are 2.143b meters from the restaurant, since each block is b meters long, 2.143b meters would equals to 2.143b/b = 2.143 blocks. And you are 5 - 2.143 = 2.857 blocks from the restaurant.