How close is the semi circle y= sqr.root of 16-x^2 to the point (1, sqr.root 3)?
using Optimization
The equation of a circle with the center at origin and passing through (1,sqrt3)is
x^2+y^2=2^2
The distance between the point and semicircle is the difference R1-R2=4-2=2
To determine how close the semi-circle y = √(16 - x^2) is to the point (1, √3), we can use optimization.
Step 1: Calculate the distance between the point and the curve.
The distance between the point (1, √3) and any point on the curve y = √(16 - x^2) can be determined using the distance formula:
Distance = √((x1 - x2)^2 + (y1 - y2)^2)
In this case, x1 = 1, y1 = √3, and y2 = √(16 - x^2). We want to minimize the distance, so the equation becomes:
Distance = √((1 - x)^2 + (√3 - √(16 - x^2))^2)
Step 2: Find the derivative of the distance equation.
To find the minimum distance, we need to find the derivative of the distance equation with respect to x:
d(Distance) / dx = 0
Step 3: Solve for x.
Set the derivative equal to 0 and solve for x:
d(Distance) / dx = 0
Step 4: Calculate the minimum distance.
Substitute the value of x obtained in Step 3 into the distance equation to find the minimum distance between the point (1, √3) and the semi-circle y = √(16 - x^2).