Viscosity word problem ~

[Mitch885885]

Hello,

Question: V(t) = -t^3 - 5t^2 + 24t + 75 is the viscosity of motor oil, where t is the temperature (10 degrees celcius). What is the temperature when the viscosity is greater than 75?

I am unsure of the order of operations for this problem. Do I need to factor the equation, find the zeros, and plot the points on a graph? How do I locate on the graph, when the temperature is greater than 75?


Thank you!

 

[skeeter]

-t^3 - 5t^2 + 24t + 75 > 75
-t^3 - 5t^2 + 24t > 0
t^3 + 5t^2 - 24 t < 0
t(t^2 + 5t - 24) < 0
t(t + 8)(t - 3) < 0

the left side equals 0 when t = -8, t = 0, and t = 3

those three values break up all possible values of t into 4 regions ...

t < -8, -8 < t < 0, 0 < t < 3, and t > 3

if t < -8, the last inequality is true
if -8 < t < 0, the last inequality is false
if 0 < t < 3, the last inequality is true
if t > 3, the last inequality is false

the range of temperatures where the viscosity > 75 would be all values of t such that t < -8 and 0 < t < 3.

why is this in the calculus section?

 

[soroban]

Hello, Mitch885885!

\(\displaystyle V(t)\:=\:-t^3\,-\,5t^2\,+\,24t\,+\,75\) is the viscosity of motor oil, where \(\displaystyle t\) is the temperature.
What is the temperature when the viscosity is greater than 75?

Evidently, you didn't understand the question: "When is \(\displaystyle V(t)\) greater than 75?"

If you have the graph before you, find the parts of the graph where \(\displaystyle V\) is above 75.


We can do this algebraically.

We want: \(\displaystyle \,-t^3\,-\,5t^2\,+\,24t\,+\,75\;>\:75\)

We have: \(\displaystyle \,-t^3\,-\,5t^2\,+\,24t\;>\;0\)

Factor: \(\displaystyle \,-t(t^2\,+\,5t\,-\,24)\:>\:0\)

Factor: \(\displaystyle \,-t(t\,-\,3)(t\,+\,8)\;>\;0\)

The expression equals 0 when \(\displaystyle t\,=\,-8,\;0,\;3\)

When is the expression positive?
We have four intervals to test.

On \(\displaystyle (-\infty,\,-8)\), try \(\displaystyle t = -9:\;(+9)(-12)(-1)\,=\,positive\) . . . yes!

On \(\displaystyle (-8,\,0)\), try \(\displaystyle t = -1:\;(+1)(-4)(+7)\,=\,negative\) . . . no

On \(\displaystyle (0,\,3)\), try \(\displaystyle t\,=\,1:\;(-1)(-2)(+9)\,=\,positive\) . . . yes!

On \(\displaystyle (3,\,\infty)\), try \(\displaystyle t\,=\,4:\;(-4)(+1)(+12)\,=\,negative\) . . . no

So the expression is positive on: \(\displaystyle \,(-\infty,\,-8)\) and \(\displaystyle (0,\,3)\)


Therefore, the viscosity is greater than 75 when \(\displaystyle t\,<\,-8\) and when \(\displaystyle 0\,<\,t\,<\,3\)


Too fast for me, skeeter!