find the point p on the line y = 6x that is closest to the point (74,0). what is the least distance between…
find the point p on the line y = 6x that is closest to the point (74,0). what is the least distance between p and (74,0)? let d be the distance between the two points. what is the objective function in terms of one number, x? d = (type an expression.)
Answer
Explanation:
Step1: Recall distance formula
The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $D=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, one point is $P=(x,6x)$ (since it lies on $y = 6x$) and the other point is $(74,0)$. So $D=\sqrt{(x - 74)^2+(6x-0)^2}$.
Step2: Expand the expression
[ \begin{align*} D&=\sqrt{(x - 74)^2+(6x)^2}\ &=\sqrt{x^{2}-148x + 74^{2}+36x^{2}}\ &=\sqrt{37x^{2}-148x + 5476} \end{align*} ]
Answer:
$\sqrt{37x^{2}-148x + 5476}$