Answer:
Step-by-step explanation:
Hello
The volume of a cube is the cube of the length of one of the sides
a) [tex]V = x^3[/tex] where x is the length of the sides for such cube:
if we differentiate we get : [tex]\frac{dV}{dx} = 3*x^2[/tex], now if we want to find the maximum error we approximate [tex]\frac{dV}{dx} [/tex] to [tex]\frac{\Delta V}{\Delta x}[/tex] which is a rude approximation.
now if we solve for [tex]\Delta V[/tex] we get: [tex]\Delta V = 3x^2*\Delta x[/tex]
now we replace x with 30 and [tex]\Delta x[/tex] with 0.4 and we get [tex]\Delta V = 3*(30)^2*0.4 = 1080[/tex]cm^3.
the relative error is given by dividing our maximum error by the volume V: [tex]\frac{1080}{30^3}= 0.0400[/tex]
the percentage error will be the product of this value with 100 (in order to get a percentage) [tex]0.0400*100 = 4\%[/tex]
So having this value, the volume of the cube will be 27000cm³ [tex]\pm[/tex] 1080 cm³
b)
The area is six times the area of a single square:
[tex]A = 6*x^2[/tex] where x is the length of the sides for such cube:
if we differentiate we get : [tex]\frac{dA}{dx} = 6*(2)*x = 12x[/tex], now if we want to find the maximum error we approximate [tex]\frac{dA}{dx} [/tex] to [tex]\frac{\Delta A}{\Delta x}[/tex] which is a rude approximation.
now if we solve for [tex]\Delta A[/tex] we get: [tex]\Delta A = 12x*\Delta x[/tex]
now we replace x with 30 and [tex]\Delta x[/tex] with 0.4 and we get [tex]\Delta A = 12*(30)*0.4 = 144[/tex]cm^2.
the relative error is given by dividing our maximum error by the surface area A: [tex]\frac{144}{6*30^2} = 0.8[/tex]
the percentage error will be the product of this value with 100 (in order to get a percentage) [tex]0.8*100 = 80\%[/tex]
Therefore, the area will be 5400cm^2 [tex]\pm[/tex] 144cm^2
