Question

calc help
Get an equation for the radius of the red balloon as a function of time.
(Hint: Drag the slider to t=1, and then t=2, and then t=3, etc, noting
each time what the radius of the Red Balloon is. See if you can spot the
pattern, and come up with an equation. (the equation will look like:
� = ___ �)

Answer 1

r=_t
@Vocaloid

Answer 2

well just try plugging in several values of t and see what sort of output values you get

Answer 3

the radius increases by .2 each time so .2t

Answer 4

Write an equation for the Volume of a sphere as a function of time.
Remember: The Volume of a Sphere is given by: � = !
!
��!
(i.e. substitute your answer from the previous question in for “�”)

Answer 5

so r is .2

Answer 6

I guess v(t)=4/3pi.2^3

Answer 7

r = 0.2t as determined by the previous step

V = (4/3)pi * r^3

so you can just plug r = 0.2t into the volume equation

Answer 8

since it asks for V in terms of t, your equation should have t in it

Answer 9

this may be easier to read

Answer 10

okay

Answer 11

so the rate of change question wants the derivative of my equation

Answer 12

yup

Answer 13

isn't the derivative of a sphere like 4pi r ^2 or something

Answer 14

it wants it in terms of t not r

from step b) you should have

V = (4/3)pi * (0.2t)^3

take the derivative of this, with respect to t, to get c)

Answer 15

so dt/dv?

Answer 16

dV/dt

Answer 17

oh crud the rate changes after t=3

Answer 18

huh. weird.

Answer 19

nevermind

Answer 20

it doesn't thank god

Answer 21

2.513t^2

Answer 22

uh I think I got something a bit diff.

V = (4/3)pi * (0.2t)^3

dV/dt = (4/3)(pi)(3)(0.2^3)t^2

Answer 23

I'd leave it in terms of pi

Answer 24

it's already in terms of V and t so just take the derivative

Answer 25

then for e) set the two V'(t) equations equal to each other and solve for t

Answer 26

is it dv/dt again?

Answer 27

I got 10

Answer 28

yes, good

Answer 29

e
blue
10
20
30
40 etc

red
.03
.27
.9
2.14
4.19
7.24
11.49
17.16
24.43
33.51

Answer 30

you just need to solve dV/dt red = dV/dt blue

dV/dt red = (4/3)(pi)(3)(0.2^3)t^2 = 0.032pi * t^2
then just set that equal to 10 and solve for t.

Answer 31

so plug in 10 at the equal sign?

Answer 32

dV/dt red = 0.032pi * t^2
dV/dt blue = 10

therefore 0.032pi * t^2 = 10

Answer 33

is that it ?

Answer 34

g is last question

Answer 35

like I said you need to solve for t.

Answer 36

oh crap okay

Answer 37

for g, you simply create two volume equations

green and orange will both be
V = (4/3)pi*r^3

then plug in the given expression for r into each equation, to generate two separate equations for V in terms of t, one for each balloon

Answer 38

t is about 10 it's not 9.8 let me see if it is 9.9

Answer 39

then set the derivatives equal to each other

Answer 40

I end up w/ t≈9.97356 for e

Answer 41

what did you do to get your answer 9.9 etc?

Answer 42

0.032pi * t^2 = 10

t^2 = 10/0.032pi

t = sqrt(10/0.032pi)

Answer 43

green (4/3)pi*10+5t^3?

Answer 44

careful with your parentheses

(4/3)pi*[10+5t]^3?

Answer 45

orange same idea with[]?

Answer 46

so what volume would the 2 increase at the same rate

Answer 47

\(\color{#0cbb34}{\text{Originally Posted by}}\) @Vocaloid
then set the derivatives equal to each other
\(\color{#0cbb34}{\text{End of Quote}}\)

Answer 48

set the derivatives equal to each other and solve for t

Answer 49

so are the equations I made the derivative or do I have t find it? slightly confused

Answer 50

the equations the volume equations in terms of t. you will then take the derivatives.

Answer 51

20pi/3

Answer 52

for green

Answer 53

gotta go in a bit but I hope I am doing it right

Answer 54

check your work again

V = (4/3)pi*(10+5t)^3
apply chain rule

dV/dt = (4/3)(pi)*3*(10+5t)^2 * (5)

Answer 55

200piy(10+5t)

Answer 56

I am lost and being rushed so I might have to do this later

Answer 57

sure

Answer 58

I have a few minutes if you could help me at least get the green side

Answer 59

@Vocaloid

Answer 60

that is the green one

if you're talking about the orange one

V(t) = (4/3)pi*(2+t^2)^3
chain rule

V'(t) = (4/3)(pi)(3)(2+t^2)^2 * 2t

Answer 61

at that point you just have to set the two derivatives equal to get

(4/3)(pi)(3)(2+t^2)^2 * 2t = (4/3)(pi)*3*(10+5t)^2 * (5), solve for t

Answer 62

?

Answer 63

@Vocaloid

Answer 64

@Hero

Answer 65

@Vocaloid

Answer 66

the pic above is my answer is that correct

Answer 67

No, remember you are taking the derivative of both volume expressions

Answer 68

20pix(10+5t)^2 for green

Answer 69

Yes

Answer 70

really ?

Answer 71

Yes

Now set it equal to the orange balloon derivative

Answer 72

I have 8pitx(2+t^2)^2 for orange

Answer 73

Good

Answer 74

is that it?

Answer 75

Again you have to set them equal to each other and solve for t

Answer 76

how do I do that like combine them like basic algebra?

Answer 77

Uh
Solving it manually is kind of a pain because you have to expand everything and factor

If possible I would recommend using a calculator

Answer 78

I literally got an error

Answer 79

I need this assignment done today could you just set the euation up for I can calculate it in terms of t

Answer 80

@Vocaloid

Answer 81

?

Answer 82

20pi*(10+5t)^2 = (4/3)(pi)(3)(2+t^2)^2 * 2t

t≈4.7408