one way to resize a ring is to fill a bar into the ring.suppose a ring that is 20 mm in diameter has to be resized to fit a fingure 16 mm n diameter
what is the length of the bar that should be inserted in order to make the ring fit the fingure? ( Hint: write an equation of the ring assuming it is centered at the origin. Determine what the y-coordinate of the bar must be and then substitute this coordinate into the equation to find x
To resize a ring by inserting a bar, we can follow these steps:
Step 1: Write an Equation of the Ring
Assuming the ring is centered at the origin, we can write the equation of the ring in terms of its diameter. The equation of a circle centered at the origin is given by: x^2 + y^2 = r^2, where r is the radius of the circle.
In this case, the given ring has a diameter of 20 mm. The radius (r) can be obtained by dividing the diameter by 2: r = 20 mm / 2 = 10 mm.
So, the equation of the ring is: x^2 + y^2 = 100.
Step 2: Determine the y-Coordinate of the Bar
In order to resize the ring to fit a finger with a diameter of 16 mm, we need to determine the y-coordinate of the bar that should be inserted.
Since the diameter of the finger is smaller than the diameter of the ring, the y-coordinate of the bar should be such that the y-value of the bar lies on the circumference of the resized ring.
For the resized ring, the new radius (r_new) can be obtained by dividing the diameter of the finger by 2: r_new = 16 mm / 2 = 8 mm.
Substituting this new radius into the equation of the ring, we get: x^2 + y^2 = 64.
Simplifying the equation, we have: y^2 = 64 - x^2.
Step 3: Substitute y-Coordinate into the Equations
To find the x-coordinate of the bar, we substitute the y-coordinate into the equation of the resized ring.
Using the equation y^2 = 64 - x^2, we substitute y = 10 into this equation. Note that the value of y is 10 because it lies on the circumference of the original ring.
Substituting y = 10, we get: (10)^2 = 64 - x^2.
Simplifying, we have: 100 = 64 - x^2.
Rearranging the equation, we get: x^2 = 64 - 100.
Simplifying further, we have: x^2 = -36.
Since x^2 cannot be negative, this means that the equation has no real solutions. Therefore, it is not possible to resize the ring to fit a finger with a diameter of 16 mm by inserting a bar.
In summary, the length of the bar that should be inserted to resize the ring cannot be determined because the equation does not yield real solutions.