**Solution:**

To find the maximum error in the measurement of length, we need to understand the relationship between mass, density, and length.

**Relationship between mass, density, and length:**

The formula for density is given by:

Density = Mass/Volume

And the formula for volume of a cube is given by:

Volume = Length³

From these formulas, we can deduce the following relationship:

Density = Mass/(Length³)

Taking the derivative with respect to each variable, we have:

d(Density)/d(Mass) = 1/(Length³)

d(Density)/d(Length) = -3*Mass/Length⁴

**Error propagation formula:**

The error propagation formula relates the maximum error in one variable to the maximum error in another variable. It is given by:

ΔY/Y = √( (∆X/X)² + (∆Z/Z)² )

Where ΔY is the maximum error in the variable Y, ∆X is the maximum error in the variable X, and ∆Z is the maximum error in the variable Z.

**Calculating the maximum error in length:**

Given that the maximum error in mass is 3% and the maximum error in density is 9%, we can use the error propagation formula to find the maximum error in length.

Let's denote the maximum error in length as ΔLength.

From the error propagation formula, we have:

(ΔDensity/Density)² = (ΔMass/Mass)² + (ΔLength/Length)²

Substituting the values, we have:

(0.09)² = (0.03)² + (ΔLength/Length)²

Simplifying the equation, we get:

0.0081 = 0.0009 + (ΔLength/Length)²

(ΔLength/Length)² = 0.0081 - 0.0009

(ΔLength/Length)² = 0.0072

Taking the square root of both sides, we have:

ΔLength/Length = √0.0072

ΔLength/Length ≈ 0.0849

Multiplying both sides by Length, we get:

ΔLength ≈ 0.0849 * Length

Therefore, the maximum error in the measurement of length is approximately 0.0849 times the length of the cube.

**Conclusion:**

The maximum error in the measurement of length of the cube is approximately 0.0849 times the length of the cube. This means that if the length of the cube is L, the maximum error in the measurement of length would be 0.0849 * L.

Rho= mass / volume
v= m/rho
a^3 = m/p
3da= dm+dp
da= 9+3/3=4
error in the measurement of length is 4%