Use differentials to approximate the change in the volume of a cube when the side is decreased from 8 to 7.99 cm. (in cm^3)
Thank you so much!!
V=x³
dV/dx = 3x²
ΔV =3x²Δx (approx.)
= 3*8²*(7.99-8)
= -1.92 cm³
To approximate the change in volume of a cube using differentials, we can use the concept of differentials or derivatives.
The volume of a cube is given by the formula V = s^3, where V is the volume and s is the length of a side.
To find the change in volume, we need to calculate the derivative of the volume function with respect to the side length.
Differentiating the volume function with respect to s, we get:
dV/ds = 3s^2
Now we can use the differential approximation formula to calculate the change in volume:
dV = (dV/ds) * ds
Given that the initial side length is 8 cm and the change in side length is 7.99 cm - 8 cm = -0.01 cm, we substitute these values into the formula:
dV = (3(8)^2) * (-0.01)
Simplifying this, we get:
dV = 3(64) * (-0.01) = -1.92 cm^3
Therefore, the change in volume of the cube when the side length is decreased from 8 to 7.99 cm is approximately -1.92 cm^3.