M propto R ^ 3 and M = 40 when R = 4 Find:a the equation connecting M and R
b M when R = 10
c R when M = 2 * 56
a) The equation connecting M and R can be found using the proportionality constant. Since M is directly proportional to R^3, we can write the equation as:
M = k * R^3
where k is the proportionality constant. We can find the value of k by substituting the given values of M and R into the equation:
40 = k * 4^3
Simplifying the equation:
40 = k * 64
Dividing both sides by 64:
k = 40/64
k = 0.625
Therefore, the equation connecting M and R is:
M = 0.625 * R^3
b) To find M when R = 10, we can substitute R = 10 into the equation we found in part a):
M = 0.625 * 10^3
Simplifying the equation:
M = 0.625 * 1000
M = 625
Therefore, when R = 10, M = 625.
c) To find R when M = 2 * 56, we can substitute M = 2 * 56 into the equation we found in part a) and solve for R:
2 * 56 = 0.625 * R^3
112 = 0.625 * R^3
Dividing both sides by 0.625:
R^3 = 112 / 0.625
R^3 = 179.2
Taking the cube root of both sides:
R = (179.2)^(1/3)
R ≈ 5.58
Therefore, when M = 2 * 56, R ≈ 5.58.