To find the distance between Alice's point (3, 4) and the magical path defined by the parametric equations, we can use the distance formula and substitute the given coordinates and parametric equations.
Alice is standing at the point (3, 4) in her neighborhood and notices a magical path defined by parametric equations: x=t^2+1 and y=t+2. To find the distance between Alice's point and the magical path, we can use the formula for the distance between two points (x1, y1) and (x2, y2) which is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Substituting the given coordinates and the parametric equations:
d = sqrt((t^2+1 - 3)^2 + (t+2 - 4)^2)
Simplifying the equation further, we get:
d = sqrt(t^2 - 4t + 4 + t^2 - 2t + 1)
d = sqrt(2t^2 - 6t + 9)
The probable question may be:
Alice is on a delightful journey through her neighborhood. The point where she is standing is represented by the coordinates (3, 4), and she notices a magical path defined by parametric equations. The x-coordinate of the path is given by x=t^2+1, and the y-coordinate is y=t+2, where t is a parameter.
Curious about the distance between her point and the magical path, Alice wonders: How far is she from this enchanting trail?
Additional Information:
Alice loves exploring the beauty of mathematics hidden in everyday adventures. The point (3, 4) represents a cozy spot where Alice often enjoys the sunset. The parametric equations describe a whimsical path weaving through the neighborhood, capturing the essence of mathematical elegance in the simplest terms. Can you help Alice find the distance between her and the magical path as she embraces the wonders of mathematics in her leisurely stroll?