The mass of the particles that a river can transport is proportional to the sixth power of the speed of the river. A certain river normally flows at a speed of 3 miles per hour. What must its speed be in order to transport particles that are 200 times as massive as usual? Round your answer to the nearest hundredth.
3 * 200^(1/6)
Let's denote the speed of the river as x miles per hour and the mass of the particles as m. We are given that the mass of the particles is proportional to the sixth power of the speed of the river.
This can be written as:
m ∝ x^6
We are told that a certain river normally flows at a speed of 3 miles per hour. Let's denote the mass of the normal particles as m0 and the corresponding speed as x0.
m0 ∝ x0^6
Given that the particles we want to transport are 200 times as massive as usual, we can write:
m = 200 * m0
Substituting m0 ∝ x0^6, we have:
m = 200 * x0^6
Since m ∝ x^6, we can write:
x^6 = (200 * x0^6) / m0
Now we can substitute the known values: x0 = 3 miles per hour, m0 = mass of normal particles, m = 200 * m0.
x^6 = (200 * (3^6)) / m0
Simplifying:
x^6 = (200 * 729) / m0
Now we want to solve for x, so we take the sixth root of both sides:
x = ((200 * 729) / m0)^(1/6)
Rounding to the nearest hundredth, we plug in the given values to calculate the speed of the river required in order to transport particles that are 200 times as massive as usual.
To find the required speed of the river, we can use the principle that the mass of the particles transported is proportional to the sixth power of the speed.
Let's denote the original speed of the river as v_1 (3 miles per hour in this case) and the mass of the particles normally transported as m_1. We are given that the mass of the particles we want to transport is 200 times m_1.
Therefore, we have the following equation representing the proportionality relationship:
m_1 ∝ v_1^6
Now, we need to find the new speed of the river, which we'll denote as v_2, and the new mass of the particles, which we'll denote as m_2. We want to find v_2, given that m_2 is 200 times m_1.
m_2 = 200 * m_1
We can rewrite the formula for the proportionality in terms of v_2 and m_2:
m_2 ∝ v_2^6
Since the two proportionality equations are equal, we can set them equal to each other:
v_1^6 = v_2^6
Now, we can solve for v_2 by taking the sixth root of both sides:
v_2 = v_1 * (m_2 / m_1)^(1/6)
Substituting the given values:
v_2 = 3 * (200 / 1)^(1/6)
Simplifying:
v_2 = 3 * 200^(1/6)
To round our answer to the nearest hundredth, we can use a calculator or online tool to calculate:
v_2 ≈ 3 * 1.43489
v_2 ≈ 4.30467
Therefore, the speed of the river must be approximately 4.30 miles per hour to transport particles that are 200 times as massive as usual.