Final answer:
Using differentials, the approximate amount of paint required to coat a cube with a 0.04 cm thick layer on all faces is 78 cm³. This is obtained by finding the difference in surface area before and after painting and considering the volume of paint as the product of the area difference and paint thickness.
Step-by-step explanation:
Approximate Amount of Paint Required
The student is tasked with applying a coat of paint to a cube with an edge length of 27 cm.
To calculate the amount of paint needed, we can use differentials.
First, calculate the surface area of the cube before painting: A = 6s2, where s is the length of an edge of the cube. For a cube with an edge of 27 cm, the surface area is A = 6(27 cm)2 = 4374 cm2.
Next, determine the surface area of the cube after a coat of paint is applied. The new edge length will be s + 2t, where t is the thickness of the paint.
Here, t = 0.04 cm. The new edge length is 27 cm + 2(0.04 cm) = 27.08 cm. So, the new surface area is A' = 6(27.08 cm)2.
To find the amount of paint needed, we calculate the difference in surface area, which gives the volume of paint required:
dV = A' - A = 6(27.08 cm)2 - 4374 cm2
Use differentials: dA = 12s ds, where ds is the change in the edge length (which is twice the paint thickness since the paint goes on all sides).
dA = 12(27 cm)(0.04 cm)
= 12.96 cm2 per face, and since there are 6 faces:
dV = 6(12.96 cm2)
= 77.76 cm3.
Rounding to the nearest cubic centimeter, the approximate amount of paint required for the job is 78 cm3.