Implicit Function: Box Dimensions & Rates of Change

[Physicsnoob90]

Homework Statement

The length ℓ, width w, and height h of a box change with time. At a certain instant the dimensions are ℓ = 4 m and
w = h = 9 m, and ℓ and w are increasing at a rate of 1 m/s while his decreasing at a rate of 6 m/s. At that instant find the rates at which the following quantities are changing.

(A) The Volume
(B) The Surface Area
(C) the length of a diagonal (round two decimals places)

Homework Equations

The Chain rule, Partial Derivative

The Attempt at a Solution

I already found A (ans: -99 m^3/s) and B (ans: -94 m^2/s)
for C:
1. i differentiated the formula L^2= ℓ^2+w^2+h^2 to 2L(dL/dt) = 2ℓ(dℓ/dt) + 2w(dw/dt) + 2h (dh/dt)
2. let dℓ/dt = dw/dt = 1 m/s and dh/dt = -6 m/s
3. my answer came as 2L(dL/dt) = 82 but I'm completely lost after this part

 

[SteamKing]

Homework Statement

The length ℓ, width w, and height h of a box change with time. At a certain instant the dimensions are ℓ = 4 m and
w = h = 9 m, and ℓ and w are increasing at a rate of 1 m/s while his decreasing at a rate of 6 m/s. At that instant find the rates at which the following quantities are changing.

(A) The Volume
(B) The Surface Area
(C) the length of a diagonal (round two decimals places)

Homework Equations

The Chain rule, Partial Derivative

The Attempt at a Solution

I already found A (ans: -99 m^3/s) and B (ans: -94 m^2/s)
for C:
1. i differentiated the formula L^2= ℓ^2+w^2+h^2 to 2L(dL/dt) = 2ℓ(dℓ/dt) + 2w(dw/dt) + 2h (dh/dt)
2. let dℓ/dt = dw/dt = 1 m/s and dh/dt = -6 m/s
3. my answer came as 2L(dL/dt) = 82 but I'm completely lost after this part

At time t, you know what the values of w, h, and l are. You want to solve 2L (dL/dt) = 82 for dL/dt.

 

[Physicsnoob90]

At time t, you know what the values of w, h, and l are. You want to solve 2L (dL/dt) = 82 for dL/dt.

would i be able to find L by square rooting (w,h,ℓ) and then multiplying it with the 2?

update: i manage to figure out the equation by doing just that. Thanks for your help!

 

[HallsofIvy]

You could but if you implicit differentiation you shouldn't! The length of the diagonal is given by L= (ℓ^2+ w^2+ h^2)^{1/2}.