Question

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 4 miles per hour. what must its speed be in order to transport particles that are 16 times as massive as usual? round your answer to the nearest hundredth.
the speed of the river must be about miles per hour.

Explanation:

Step1: Set up the proportionality equation

Let $m$ be the mass of the particles and $v$ be the speed of the river. We have $m = kv^{6}$, where $k$ is the constant of proportionality. Initially, when $v = 4$, $m_1=k\times4^{6}$. Let the new speed be $v_2$ and the new mass $m_2 = 16m_1$. Then $m_2=k\times v_2^{6}$.

Step2: Substitute $m_2 = 16m_1$ into the equations

Since $m_1 = k\times4^{6}$ and $m_2=k\times v_2^{6}$, and $m_2 = 16m_1$, we get $k\times v_2^{6}=16\times k\times4^{6}$. Since $k
eq0$, we can cancel out $k$ from both sides of the equation, and we have $v_2^{6}=16\times4^{6}$.

Step3: Simplify the right - hand side

We know that $16 = 2^{4}$ and $4^{6}=(2^{2})^{6}=2^{12}$, so $16\times4^{6}=2^{4}\times2^{12}=2^{16}$. Then $v_2^{6}=2^{16}$.

Step4: Solve for $v_2$

Take the sixth - root of both sides: $v_2 = 2^{\frac{16}{6}}=2^{\frac{8}{3}}=\sqrt[3]{2^{8}}=\sqrt[3]{256}\approx6.35$.

Answer:

$6.35$