A contact lens has a spherical shape and pretend it is being held on a finger ready to put into an eye. (All measures are in millimetres.) Suppose we take the semi-circular cross-section through the middle of the lens. Find the equation of the lens if the centrepoint of the cross-section is (0,3) and a speck of dirt on the lens at (3,3)Find the radius. Round off your answer to the nearest decimal place.
Then find the height of a point that is 2 mm from the centre. Round off your answer to the nearest decimal place.
To find the equation of the lens, we need to determine its shape. Since the contact lens has a spherical shape, we can use the equation of a circle to find its radius.
1. Determine the radius:
Given that the center point of the cross-section is (0,3) and a speck of dirt on the lens is located at (3,3), we can use the distance formula to find the radius.
Distance formula: d = √((x2 - x1)^2 + (y2 - y1)^2)
Using the distance formula, we have:
d = √((3 - 0)^2 + (3 - 3)^2)
d = √(9 + 0)
d = √9
d = 3
So, the radius of the lens is 3.
2. Find the height of a point 2 mm from the center:
To find the height of a point 2 mm from the center, we need to consider the radius and apply a bit of trigonometry.
Since the lens has a spherical shape, we know that the cross-section is a semicircle. Hence, the height of the point can be determined using the Pythagorean theorem.
Let's consider a right triangle formed by the center of the cross-section (0,3), the point on the circumference (0,r), and the desired point (0,3+2).
Using the Pythagorean theorem, we have:
(r+2)^2 = r^2 + 2^2
Simplifying the equation, we get:
r^2 + 4r + 4 = r^2 + 4
Canceling out the r^2 terms, we get:
4r = 0
Solving for r, we find:
r = 0
So, the height of a point 2 mm from the center is 0 mm.
In summary:
- The radius of the lens is 3 mm.
- The height of a point 2 mm from the center is 0 mm.