Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
To find the point [tex]\( P \)[/tex] on the graph of the function [tex]\( y = \sqrt{x} \)[/tex] that is closest to the point [tex]\( (3, 0) \)[/tex], we need to minimize the distance between any point on the curve [tex]\( (x, \sqrt{x}) \)[/tex] and the point [tex]\( (3, 0) \)[/tex].
Let's denote the distance between a point [tex]\( (x, \sqrt{x}) \)[/tex] on the curve and the point [tex]\( (3, 0) \)[/tex] as [tex]\( D \)[/tex].
First, we write the distance formula [tex]\( D \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[ D = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} \][/tex]
Substituting [tex]\( (x_1, y_1) = (x, \sqrt{x}) \)[/tex] and [tex]\( (x_2, y_2) = (3, 0) \)[/tex], we get:
[tex]\[ D = \sqrt{(x - 3)^2 + (\sqrt{x} - 0)^2} \][/tex]
Simplify the expression:
[tex]\[ D = \sqrt{(x - 3)^2 + (\sqrt{x})^2} \][/tex]
Since [tex]\((\sqrt{x})^2 = x\)[/tex], we can rewrite [tex]\( D \)[/tex] as:
[tex]\[ D = \sqrt{(x - 3)^2 + x} \][/tex]
To find the minimum distance, we need to find the [tex]\( x \)[/tex] value that minimizes this distance function. Through optimization techniques and calculations, the [tex]\( x \)[/tex]-coordinate of point [tex]\( P \)[/tex] is found to be:
[tex]\[ x = 2.4999999114637803 \][/tex]
Therefore, the [tex]\( x \)[/tex]-coordinate of the point [tex]\( P \)[/tex] on the graph of [tex]\( y = \sqrt{x} \)[/tex] that is closest to the point [tex]\( (3, 0) \)[/tex] is approximately:
[tex]\[ x \approx 2.5 \][/tex]
Let's denote the distance between a point [tex]\( (x, \sqrt{x}) \)[/tex] on the curve and the point [tex]\( (3, 0) \)[/tex] as [tex]\( D \)[/tex].
First, we write the distance formula [tex]\( D \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[ D = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} \][/tex]
Substituting [tex]\( (x_1, y_1) = (x, \sqrt{x}) \)[/tex] and [tex]\( (x_2, y_2) = (3, 0) \)[/tex], we get:
[tex]\[ D = \sqrt{(x - 3)^2 + (\sqrt{x} - 0)^2} \][/tex]
Simplify the expression:
[tex]\[ D = \sqrt{(x - 3)^2 + (\sqrt{x})^2} \][/tex]
Since [tex]\((\sqrt{x})^2 = x\)[/tex], we can rewrite [tex]\( D \)[/tex] as:
[tex]\[ D = \sqrt{(x - 3)^2 + x} \][/tex]
To find the minimum distance, we need to find the [tex]\( x \)[/tex] value that minimizes this distance function. Through optimization techniques and calculations, the [tex]\( x \)[/tex]-coordinate of point [tex]\( P \)[/tex] is found to be:
[tex]\[ x = 2.4999999114637803 \][/tex]
Therefore, the [tex]\( x \)[/tex]-coordinate of the point [tex]\( P \)[/tex] on the graph of [tex]\( y = \sqrt{x} \)[/tex] that is closest to the point [tex]\( (3, 0) \)[/tex] is approximately:
[tex]\[ x \approx 2.5 \][/tex]
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.