Question

The length ℓ, width w, and height h of a box change with time. At a certain instant the dimensions are ℓ = 7 m and w = h = 2 m, and ℓ and w are increasing at a rate of 3 m/s while h is decreasing at a rate of 1 m/s. At that instant find the rates at which the following quantities are changing. The length of a diagonal. (Round your answer to two decimal places.)

Answer 1

the main diagonal z can be found using
z^2 = ℓ^2 + w^2 + h^2
z dz/dt = ℓ dℓ/dt + w dw/dt h dh/dt
So, plug in your numbers and solve for dz/dt

If it's some other diagonal you want, then solve using only two dimensions.

Answer 2

d^2 = ℓ^2 + w^2 + h^2

differentiating ... 2 d dd/dt = 2 ℓ dℓ/dt + 2 w dw/dt + 2 h dh/dt

divide by 2 and substitute ... √57 dd/dt = (7 * 3) + (2 * 3) + (2 * -1)

solve for dd/dt