Find the point on the line [tex]y=8x+4[/tex] closest to the point [tex](0,-6)[/tex].
The function giving the distance between the point and the line is [tex]s=[/tex] $\square$ (Enter a function of [tex]x[/tex])
Answer (2)
Define the distance function s ( x ) between a point ( x , 8 x + 4 ) on the line and the point ( 0 , − 6 ) .
Express the distance function as s ( x ) = x 2 + ( 8 x + 10 ) 2 .
Simplify the expression to obtain s ( x ) = 65 x 2 + 160 x + 100 .
The function giving the distance between the point and the line is 65 x 2 + 160 x + 100 .
Explanation
Problem Setup We are given the line y = 8 x + 4 and the point ( 0 , − 6 ) . Our goal is to find the point on the line that is closest to the given point. This involves minimizing the distance between a general point ( x , y ) on the line and the point ( 0 , − 6 ) .
Expressing Distance as a Function of x The distance s between the point ( x , y ) on the line and the point ( 0 , − 6 ) is given by the distance formula: s = ( x − 0 ) 2 + ( y − ( − 6 ) ) 2 = x 2 + ( y + 6 ) 2 Since y = 8 x + 4 , we can substitute this into the distance formula to express s as a function of x :
s ( x ) = x 2 + ( 8 x + 4 + 6 ) 2 = x 2 + ( 8 x + 10 ) 2 s ( x ) = x 2 + ( 64 x 2 + 160 x + 100 ) = 65 x 2 + 160 x + 100
Distance Function The function giving the distance between the point and the line is: s ( x ) = 65 x 2 + 160 x + 100
Examples
In manufacturing, you might need to find the closest point on a conveyor belt (represented by a line) to a robotic arm's fixed position. Minimizing this distance ensures the robot can efficiently pick up items from the belt. This problem demonstrates how to optimize distances in real-world scenarios, improving efficiency and reducing costs.
The function giving the distance between the point and the line is s ( x ) = 65 x 2 + 160 x + 100 . This distance function reflects how far a point on the line is from the point ( 0 , − 6 ) . To find the closest point, we would minimize this distance function with respect to x .
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