Question

Suppose that the amount of air in a balloon after t hours is given by V(t)=t^3 -6t^2 + 35 Estimate the instantaneous rate of change of the volume after 5 hours.

Answer 1

Answer: Hi hope this helps

Answer 2

The $\sf ins ta ntan eous$ rate of change of the volume after 5 hours is 15

Given :

The amount of air in a balloon after t hours is given by V(t) = t³- 6t² + 35

To find :

Estimate the $\sf ins ta ntan eous$ rate of change of the volume after 5 hours.

Solution :

Step 1 of 3 :

Write down the the equation

Here it is given that the amount of air in a balloon after t hours is given by

V(t) = t³- 6t² + 35

Step 2 of 3 :

Find $\sf ins ta ntan eous$ rate of change of the volume

V(t) = t³- 6t² + 35

Differentiating both sides we get

V'(t) = 3t² - 6 × 2t + 0

⇒ V'(t) = 3t² - 12t

Which is $\sf ins ta ntan eous$ rate of change of the volume at any time t

Step 3 of 3 :

Find $\sf in sta ntan eous$ rate of change of the volume after 5 hours.

Putting t = 5 in V'(t) we get

⇒ V'(5) = 3 × 5² - 12 × 5

⇒ V'(5) = 3 × 25 - 60

⇒ V'(5) = 75 - 60

⇒ V'(5) = 15

Hence the $\sf in sta ntan eous$ rate of change of the volume after 5 hours is 15