SigepBrandon
New member
- Joined
- Feb 17, 2011
- Messages
- 39
The problem reads:
A rectangular box has its edges changing length as time passes. At a particular instant, the sides have lengths a=150 ft.,b=80 ft. c=50 ft. 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.
My attempt:
I wrote the formulas for L[sub:2trw5wke]a[/sub:2trw5wke], L[sub:2trw5wke]b[/sub:2trw5wke], L[sub:2trw5wke]c[/sub:2trw5wke] and for V:
L[sub:2trw5wke]a[/sub:2trw5wke]= 150+100t
L[sub:2trw5wke]b[/sub:2trw5wke]= 80-20t
L[sub:2trw5wke]c[/sub:2trw5wke]= 50+5t
V=L[sub:2trw5wke]a[/sub:2trw5wke]*L[sub:2trw5wke]b[/sub:2trw5wke]*L[sub:2trw5wke]c[/sub:2trw5wke]
Next I wrote V as a composite function of L(t) and attempted to differentiate to find dv/dt, but I couldn't keep track of everything and got lost in the product rule. So I tried to write a partial chain rule formula for dv/dt, but I could only write it as dt/dv with the information I created. The attempt is as follows:
\(\displaystyle \frac{\partial t}{\partial V}= \frac{\partial L_{a}}{\partial t}\frac{\partial t}{\partial V}+\frac{\partial L_{b}}{\partial t}\frac{\partial t}{\partial V}+\frac{\partial L_{c}}{\partial t}\frac{\partial t}{\partial V}\)
My only problem with the above formula is that is does not make sense...! I need dV/dt so... I came up with:
\(\displaystyle \frac{\partial V}{\partial t}= \frac{\partial L_{a}}{\partial V}\frac{\partial V}{\partial t}+\frac{\partial L_{b}}{\partial V}\frac{\partial V}{\partial t}+\frac{\partial L_{c}}{\partial V}\frac{\partial V}{\partial t}\)
But V is not in my L(t)'s... I either need to reevaluate my initial assumptions, or am missing a step or formula... I'm just not sure and the list could go on. Please Help! I'll keep trying to work my way through the composite angle unless someone is kind enough to save me from that torture.
thanks.
Brandon
A rectangular box has its edges changing length as time passes. At a particular instant, the sides have lengths a=150 ft.,b=80 ft. c=50 ft. 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.
My attempt:
I wrote the formulas for L[sub:2trw5wke]a[/sub:2trw5wke], L[sub:2trw5wke]b[/sub:2trw5wke], L[sub:2trw5wke]c[/sub:2trw5wke] and for V:
L[sub:2trw5wke]a[/sub:2trw5wke]= 150+100t
L[sub:2trw5wke]b[/sub:2trw5wke]= 80-20t
L[sub:2trw5wke]c[/sub:2trw5wke]= 50+5t
V=L[sub:2trw5wke]a[/sub:2trw5wke]*L[sub:2trw5wke]b[/sub:2trw5wke]*L[sub:2trw5wke]c[/sub:2trw5wke]
Next I wrote V as a composite function of L(t) and attempted to differentiate to find dv/dt, but I couldn't keep track of everything and got lost in the product rule. So I tried to write a partial chain rule formula for dv/dt, but I could only write it as dt/dv with the information I created. The attempt is as follows:
\(\displaystyle \frac{\partial t}{\partial V}= \frac{\partial L_{a}}{\partial t}\frac{\partial t}{\partial V}+\frac{\partial L_{b}}{\partial t}\frac{\partial t}{\partial V}+\frac{\partial L_{c}}{\partial t}\frac{\partial t}{\partial V}\)
My only problem with the above formula is that is does not make sense...! I need dV/dt so... I came up with:
\(\displaystyle \frac{\partial V}{\partial t}= \frac{\partial L_{a}}{\partial V}\frac{\partial V}{\partial t}+\frac{\partial L_{b}}{\partial V}\frac{\partial V}{\partial t}+\frac{\partial L_{c}}{\partial V}\frac{\partial V}{\partial t}\)
But V is not in my L(t)'s... I either need to reevaluate my initial assumptions, or am missing a step or formula... I'm just not sure and the list could go on. Please Help! I'll keep trying to work my way through the composite angle unless someone is kind enough to save me from that torture.
thanks.
Brandon