Answer:
a = 3
Step-by-step explanation:
The volume of a sphere is given by the formula V = (4/3)πr³.
To find dV/dr, we can differentiate both sides of this equation with respect to r:
[tex]\dfrac{\text{d}V}{\text{d}r}=3 \cdot \dfrac{4}{3} \pi r^{3-1}[/tex]
[tex]\dfrac{\text{d}V}{\text{d}r}=4 \pi r^{2}[/tex]
The instantaneous rate of change of the volume V of a sphere with respect to its radius r can be expressed as dV/dr = a(V/r). To find the value of a, we can substitute the expressions for dV/dr and V into the equation:
[tex]\begin{aligned}\dfrac{\text{d}V}{\text{d}r}&=a\:\dfrac{V}{r}\\\\4 \pi r^{2}&=a\:\dfrac{\dfrac{4}{3} \pi r^3}{r}\end{aligned}[/tex]
Solve for a:
[tex]\begin{aligned}4 \pi r^{2}\cdot r&=a\:\dfrac{\dfrac{4}{3} \pi r^3}{r}\cdot r\\\\4 \pi r^{3}&=a\:\dfrac{4}{3} \pi r^3\\\\3 \cdot 4\pi r^3&=a\;4\pi r^3\\\\3&=a\end{aligned}[/tex]
Therefore, the value of a is:
[tex]\huge\boxed{\boxed{a=3}}[/tex]