Solve for Contact Lens Diameter Using Depth and Radius of Curvature Formula

Question

OPTICS: A contact lens with the appropriate depth ensures proper fit and oxygen permeation. The depth of a lens can be calculated using the formula $S = r - \sqrt{r^2 - \left(\frac{d}{2}\right)^2}$, where $S$ is the depth, $r$ is the radius of curvature, and $d$ is the diameter, with all units in millimeters. a. If the depth of the contact lens is 1.15 millimeters and the radius of curvature is 7.50 millimeters, what is the diameter of the contact lens?

Accepted Answer

7.98 mm

Answer

Step 1: Substitute the known values into the given formula. We know $S=1.15$ mm and $r=7.50$ mm, so plug these into $S = r - \sqrt{r^2 - \left(\frac{d}{2}\right)^2}$: $1.15 = 7.50 - \sqrt{(7.50)^2 - \left(\frac{d}{2}\right)^2}$ Step 2: Rearrange the equation to isolate the square root term. Subtract 7.50 from both sides and multiply by -1: $\sqrt{(7.50)^2 - \left(\frac{d}{2}\right)^2} = 7.50 - 1.15 = 6.35$ Step 3: Square both sides of the equation to eliminate the square root: $(7.50)^2 - \left(\frac{d}{2}\right)^2 = (6.35)^2$ Step 4: Calculate the squared values: $56.25 - \left(\frac{d}{2}\right)^2 = 40.3225$ Step 5: Rearrange to solve for $\left(\frac{d}{2}\right)^2$: $\left(\frac{d}{2}\right)^2 = 56.25 - 40.3225 = 15.9275$ Step 6: Take the square root of both sides to find $\frac{d}{2}$: $\frac{d}{2} = \sqrt{15.9275} \approx 3.9909$ Step 7: Multiply by 2 to solve for the diameter $d$: $d \approx 3.9909 \times 2 = 7.98$ mm