Question

A balloon is at a height of 50 meters, and is rising at the constant rate of 5 m/sec. a bicyclist passes beneath it, traveling in a straight line at the constant speed of 10 m/sec. How fast is the distance between the bicyclist and the balloon increasing 2 seconds later?

Answer 1

What is the formula for the distance between the bicyclist and the balloon?

Answer 2

Oooh, fun one =)

Answer 3

I find it always helps to draw a picture..

Answer 4

Sqrt(10^2+5^2)

Answer 5

I drew a right triangle with the balloon on top and the bike on the right acute corner

Answer 6

think of it as if the balloon were stationary

Answer 7

Refer to picture attached

Answer 8

:D looks like mine.

Answer 9

11.180339887498948482045868343656

Answer 10

Write an equation for d.

Answer 11

h=sqrt(50^2+y^2)

Answer 12

=) I'm sure you're more artistic than me though.

Answer 13

d sorry, I labeled it h

Answer 14

Not quite.

Answer 15

y is one leg of the triangle, and it starts at 50, but we're not going to worry about that yet. The other leg is x.

Answer 16

oh boo.

Answer 17

after time t seconds, the cyclist will have travelled 2t meters horizontally away from the balloon. the balloon would have risen 50 + 10t meters vertically away from the cyclist. the distance between them is sqrt((2t)^2 + (50 + 10t)^2)

= sqrt(104t^2 + 2500 + 1000t)

rate of change of distance = (208*t + 1000)/2sqrt(104t^2+2500+1000t)

= (104t + 500)/sqrt(104t^2 + 2500 + 1000t)

at t = 4, rate of change = (104*4 + 500)/sqrt(104*16 + 2500 + 4000)

Answer 18

so d=sqrt(x^2+y^2)

Answer 19

Yes. Exactly.

Answer 20

ok then differentiate that?

Answer 21

for x?

Answer 22

Well, it's changing two ways. Let's start with dd/dx

Answer 23

hmm ok.

Answer 24

ok.

Answer 25

dh/dx=(x+y dy/dx)/sqrt(x^2+y^2)

Answer 26

Oh, okay. I think I have a simpler way.

Answer 27

Mind if we change our approach?

Answer 28

no worries.

Answer 29

First, write x as a function of t, time. Also write y as a function of t, time.

Answer 30

dx/dt=5 and dy/dt=10

Answer 31

I was hpoing there was a simpler way since we had that info

Answer 32

Yes.

Answer 33

Isn't it true that x = 10t?

Answer 34

yes!

Answer 35

And y?

Answer 36

oh eek. no sure.

Answer 37

What does y equal in terms of t?

Answer 38

It starts at 50.

Answer 39

And increases at a rate of 5m/s.

Answer 40

wait, I'm confused. x is height right?

Answer 41

how does it start at 50? because it passes under the balloon?

Answer 42

No, x is the horizontal distance, the distance the bike has traveled.

Answer 43

y is the vertical distance, the distance of the balloon from the ground, and it starts at 50 m high.

Answer 44

y=5t

Answer 45

+50

Answer 46

Yes. Exactly. y = 50 + 5t.

Answer 47

Okay, so x = 10t. y = 50 + 5t. Also, we got that d = sqrt(x^2 + y^2)

Answer 48

Follow?

Answer 49

so we just plug it in?

Answer 50

oh goddamn, i misread

y = 50 + 5t, x = 2t

distance = sqrt(2500 + 500t + 29t^2)

d(distance)/t = (500 + 58t)/2sqrt(2500 + 500t + 29t^2)

= (250 + 29t)/sqrt(2500 + 500t + 29t^2)

at t = 2: d(distance)/t = (250 + 58)/sqrt(2500 + 1000 + 116)

Answer 51

Yes. We substitute for x and y.

Answer 52

So d = sqrt((10t)^2 + (50 + 5t)^2)

Answer 53

ok. then do it like an mx+b problem? eek

Answer 54

No not quite.

Answer 55

It's going to be a derivative problem. But first simplify.

Answer 56

oh ok wonderful

Answer 57

I'll let you simplify that.

Answer 58

ok h=sqrt(125t^2+500t+2500)

Answer 59

Yes.

Answer 60

I'll simplify it some more. give me a sec

Answer 61

That's as far as you need to simplify it.

Answer 62

So what is the question asking for? dh/dt, right?

Answer 63

oh ok. so just take the derivative

Answer 64

Yes =)

Answer 65

k so dh/dt=(125t+250)/sqrt(125t^2+500t+2500)

Answer 66

Yes, I believe.

Answer 67

You're beautiful, btw.

Answer 68

oh thanks. how do you know which one I am? :P

Answer 69

I don't. They're both beautiful.

Answer 70

hahaha. good answer.

Answer 71

ok, so after I take the derivative, I still have t in htere as a variable. :(

Answer 72

What does the question ask?

Answer 73

oh psh, I'm a dunce. sorry

Answer 74

Post the answer, I am trying to work this also.

Answer 75

Haha don't be sorry.

Answer 76

it asks for the value of the derivative when t = 2

Answer 77

ok got it. 500/sqrt(4000)

Answer 78

About 8 meters/sec

Answer 79

yes.

Answer 80

Do you have the approved answer?

Answer 81

no, I don't

Answer 82

smoothmath for our edification is that a correct answer

Answer 83

Good work, Nasryn. =)

Answer 84

thank you!

Answer 85

It is the right answer.

Answer 86

I take it a medal is earned, thanks for the step by step and lucid explanation and drawing.

Answer 87

finally on my last one! wonderful! I might get some sleep tonight. :P

Answer 88

Haha. I appreciate it, Radar. Math is it's own reward though. =)

Answer 89

You are so right !!

Answer 90

its*

Answer 91

Yay, Nasryn! Don't forget to have fun too.

Answer 92

jesus christ, if she's the one on the right, she's smoking.

Answer 93

if she's the one on the left, still hot.

Answer 94

I'm the one on the right.

Answer 95

:P

Answer 96

yeah...don't mind the math geeks as we drool for a while

our saliva descends at 2 meters/sec downwards

Answer 97

Yeah. Freaking beautiful. Mind if I facebook you?