Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Connect with professionals ready to provide precise answers to your questions on our comprehensive Q&A platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
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]
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.