An object is placed in front of a converging lens in such a position that the lens (f = 14.0 cm) creates a real image located 27.0 cm from the lens. Then, with the object remaining in place, the lens is replaced with another converging lens (f = 18.0 cm). A new, real image is formed. What is the image distance of this new image?

I will be happy to check your optics equation work

To find the image distance of the new image formed by the second converging lens, we will use the lens formula:

1/f = 1/dₒ + 1/dᵢ

Where:
- f is the focal length of the lens
- dₒ is the object distance from the lens
- dᵢ is the image distance from the lens

In this case, we are given:
- The focal length of the first converging lens (f₁) = 14.0 cm
- The image distance created by the first lens (dᵢ₁) = 27.0 cm

Step 1: Calculate the object distance from the first lens (dₒ₁).
Since the first lens creates a real image, the object distance will be negative.
Using the lens formula, we have:
1/14 = 1/dₒ₁ + 1/27
Solving this equation will give us the object distance dₒ₁.

Step 2: Calculate the new image distance (dᵢ₂) using the lens formula.
Using the lens formula with the values:
- The focal length of the second converging lens (f₂) = 18.0 cm
- The object distance from the first lens (dₒ₁)
We have:
1/18 = 1/dₒ₁ + 1/dᵢ₂

Solving this equation will give us the image distance dᵢ₂ of the new image formed by the second converging lens.

Let's solve it step by step.

Step 1: Calculate dₒ₁:
1/14 = 1/dₒ₁ + 1/27
Rearranging the equation, we have:
1/dₒ₁ = 1/14 - 1/27
Simplifying this expression, we get:
1/dₒ₁ = (27 - 14)/(14 * 27)
Calculating further, we have:
1/dₒ₁ = 13/378
Taking the reciprocal of both sides, we get:
dₒ₁ = 378/13

Step 2: Calculate dᵢ₂:
Now that we have the object distance from the first lens (dₒ₁), we can calculate the new image distance:
1/18 = 1/dₒ₁ + 1/dᵢ₂
Substituting the value of dₒ₁, we have:
1/18 = 1/(378/13) + 1/dᵢ₂
Simplifying this expression, we get:
1/18 = 13/378 + 1/dᵢ₂
Common denominator on the right side:
1/18 = (13 + 378)/(378 * 13) + 1/dᵢ₂
Simplifying further, we have:
1/18 = 391/4914 + 1/dᵢ₂
Subtracting 391/4914 from both sides:
1/18 - 391/4914 = 1/dᵢ₂
Calculating further, we have:
(4914 - 391*18)/(18 * 4914) = 1/dᵢ₂
4760/88452 = 1/dᵢ₂
Taking the reciprocal of both sides, we get:
dᵢ₂ = 88452/4760

Therefore, the image distance of the new image formed by the second converging lens is approximately 18.58 cm.