problem: The pressure \( P \) exerted by wind on a sail varies jointly as the area \( A \) of the sail and the cube of the wind's velocity \( V \). When the velocity is \( 8 \) miles per hour, the pressure on a sail of \( 2 \) square feet is \( 4 \) pounds. Find the wind velocity when the pressure on \( 4 \) square feet of sail is \( 32 \) pounds. Let's think step by step and output the final answer within \boxed{}.

answer: 12.8

question: The pressure \( P \) exerted by wind on a sail varies jointly as the area \( A \) of the sail and the cube of the wind's velocity \( V \). When the velocity is \( 8 \) miles per hour, the pressure on a sail of \( 2 \) square feet is \( 4 \) pounds. Find the wind velocity when the pressure on \( 4 \) square feet of sail is \( 32 \) pounds. Let's think step by step and output the final answer within \boxed{}.


problem:
Given that circle $C$ passes through points $P(0,-4)$, $Q(2,0)$, and $R(3,-1)$.
$(1)$ Find the equation of circle $C$.
$(2)$ If the line $l: mx+y-1=0$ intersects circle $C$ at points $A$ and $B$, and $|AB|=4$, find the value of $m$. Let's think step by step and output the final answer within \boxed{}.

answer: \frac{4}{3}

question:
Given that circle $C$ passes through points $P(0,-4)$, $Q(2,0)$, and $R(3,-1)$.
$(1)$ Find the equation of circle $C$.
$(2)$ If the line $l: mx+y-1=0$ intersects circle $C$ at points $A$ and $B$, and $|AB|=4$, find the value of $m$. Let's think step by step and output the final answer within \boxed{}.