!!!!!!!!Use differntial to solve this question for me plz!!!!!!!!!!!?

Question

use differential to estimate the amount of paint in cubic meters required to paint the four sdies,top and bottom of a box wich is 1.5m * 1.5m * 2m if a coat of paint is 2mm think.

Answer 1

Use differential with respect to what?

1.5 * 2 = 3 sq m

3 * 4 = 12 sq m for sides

1.5 * 1.5 = 2.25 sq m

2.25 * 2 = 4.5 sq m for sides

12 + 4.5 = 16.5 sq m total surface area.

16.5 sq m * 2 mm * (1 m/1000 mm) = 16.5 * 0.002 = 0.033 cu m

Answer 2

Denote the sides fo the box a, b, c. The volume of the box is V = abc. Differential of that equals dV = bc*da+ac*db+ab*dc. Ammount of paint (=Vp) is approximately equal to dV where da = db = dc = 2*dp. dp stands for the thickness of the paint. Mathematically

Vp approx.= 2(ab+ac+bc)dp = 0.033m^3

Note that there is an area of the box 2(ab+ac+bc) multiplied by dp.

Answer 3

V = L*W*H
dV = L*WdH + W*H*dL + L*HdW
dH = dL = dW = 4 mm
dV = (4mm)(1.5*1.5 m^2 + 1.5*2 m^2 + 1.5*2 m^2)
dV = (0.004 m)(2.25 + 3 + 3) m^2
dV = (0.004)(8.25) m^3
dV = 0.033 m^3

Check:
V2 - V1 = (1.504)(1.504)(2.004) - 1.5*1.5*2
V2 - V1 = 0.033080064 m^3