the volume v of a sphere of radius r is given by v=4/3pie r^3 what is the fractional change in v for a fractional change in r
dv = 4 pi r^2 dr note(surface area * dr)
dv/v = 4 pi r^2 dr / (4/3) pi r^3 =3 dr/r
To find the fractional change in volume (ΔV) for a fractional change in radius (Δr) of a sphere, you can use differential calculus.
1. Start with the formula for the volume of a sphere:
V = (4/3)πr³
2. Differentiate both sides of the equation with respect to r:
dV/dr = (4/3)π * 3r²
= 4πr²
3. To understand the relationship between ΔV and Δr, we can divide both sides by V:
(dV/dr) / V = (4πr²) / [(4/3)πr³]
= (4πr²) / [(4π/3)r³]
= 3 / r
4. Now, let's find the fractional change in volume (ΔV/V) for a fractional change in radius (Δr/r):
ΔV/V = (dV/dr) / V * Δr/r
= (3/r) * (Δr/r)
= 3 (Δr/r²)
Therefore, the fractional change in volume for a fractional change in radius is 3 times the fractional change in radius divided by the square of the original radius.