Question

(6) (Bonus) A rectangular box has its edges changing length as time passes. At a particular instant, the sides have lengths a = 150 feet, b = 80 feet, and c = 50 feet. At that instant, a is increasing at 100 feet/sec, b is decreasing 20 feet/sec, and c is increasing at 5 feet/sec. Determine if the volume of the box is increasing, decreasing, or not changing at all, at that instant.

Answer 1

Answer:

Increasing,$310000ft^3/s$

Step-by-step explanation:

We are given that

a=150 feet

b=80 feet

c=50 feet

$\frac{da}{dt}=100ft/s$

$\frac{db}{dt}=-20ft/s$

$\frac{dc}{dt}=5ft/s$

We have to find the volume of the box  of the box increasing,decreasing or not changing at all at that instant.

We know that

Volume of box=$lbh$

Using the formula

Volume of box,V=abc

Differentiate w.r.t t

$\frac{dV}{dt}=\frac{da}{dt}bc+\frac{db}{dt}(ac)+\frac{dc}{dt}(ab)$

Substitute the values

$\frac{dV}{dt}=100(80\times 50)-20(150\times 50)+5(150\times 80)$

$\frac{dV}{dt}=310000ft^3/s$

Hence, the volume of the box is increasing .