Given: error in measuring radius = 2%, error in measuring mass = 3%

To find: error in measuring density of the material of the sphere

Formula: Density = Mass / (4/3 * π * r^3)

Let the measured values of radius and mass be r and m respectively, and the measured density be ρ.

Error in measuring radius = 2% of r = 0.02r
Error in measuring mass = 3% of m = 0.03m

Using the formula for density, we have:

ρ = m / (4/3 * π * r^3)

Differentiating both sides with respect to each of the variables m and r, we get:

dρ/dm = 1 / (4/3 * π * r^3)
dρ/dr = -3m / (4/3 * π * r^4)

Error in measuring density due to error in measuring mass:

δρ/ρ = (dρ/dm) * (δm/m)
= (1 / (4/3 * π * r^3)) * (0.03m/m)
= 3 / (4πr^3) * 0.03

Error in measuring density due to error in measuring radius:

δρ/ρ = (dρ/dr) * (δr/r)
= (-3m / (4/3 * π * r^4)) * (0.02r/r)
= -6 / (4πr^4) * m * 0.02

Total error in measuring density:

δρ/ρ = 3 / (4πr^3) * 0.03 - 6 / (4πr^4) * m * 0.02
= 0.09 / r^3 - 0.12m / r^4

Substituting the value of r^3 in terms of m from the formula for density, we get:

ρ = m / (4/3 * π * r^3)
r^3 = m / (4/3 * π * ρ)
Substituting these values in the expression for error in density, we get:

δρ/ρ = 0.09 * (4/3 * π * ρ/m)^(1/3) - 0.12 * m / (m^(1/3) * 4/3 * π * ρ^(4/3))

Simplifying this expression, we get:

δρ/ρ = 0.09 * (3/4π)^(1/3) * (ρ/m)^(1/3) - 0.09 * (3/4π)^(1/3)

Substituting the given values, we get:

δρ/ρ = 0.09 * (3/4π)^(1/3) * (ρ/4m/3)^(1/3) - 0.09 * (3/4π)^(1/3)

Simplifying this expression, we get:

δρ/ρ = 0.09 * (3/4π)^(1/3) * (3/4)^(1/3) - 0.09 * (3/4π)^(1/3)

δρ/ρ =