The mass of a spherical ball varies as the cube of the radius and inversely as the surface area . when the radius is 14, area =821 , the mass is 59.4
Find
(A) the equation connecting the relationship
(B) mass when Area = 1018 , radius = 9cm
Let's start by defining the variables:
m = mass
r = radius
A = surface area
Since the mass varies as the cube of the radius, we can write the equation as: m = kr^3. (Equation 1)
Since the mass varies inversely as the surface area, we can write the equation as: m = k/A. (Equation 2)
Given that when the radius is 14, the area is 821 and the mass is 59.4, we can substitute these values into the equations to find the value of k.
From Equation 1:
59.4 = k * 14^3
59.4 = k * 2744
k = 59.4/2744
k ≈ 0.02165414
From Equation 2:
59.4 = 0.02165414/A
Now, let's find the equation connecting the relationship between mass (m), radius (r), and surface area (A).
From Equation 1, we know that m = kr^3. We can solve this equation for r^3:
r^3 = m/k. (Equation 3)
From Equation 2, we know that m = k/A. We can solve this equation for A:
A = k/m. (Equation 4)
Since we need an equation connecting mass, radius, and surface area, we can substitute Equation 4 into Equation 3:
r^3 = (k/m) * m
r^3 = k
Therefore, the equation connecting the relationship is r^3 = k.
Now, let's calculate the mass when the area is 1018 and the radius is 9cm.
From Equation 4:
A = k/m
1018 = 0.02165414/m
Now let's solve for m:
m = 0.02165414/1018
m ≈ 0.0000213 kg
So, the mass when the area is 1018 and the radius is 9cm is approximately 0.0000213 kg.