Question

A cheap cell phone camera uses a single lens to form an image on a sensor that is 10 mm high and 4.9 mm behind the lens. Ignore the tilting that occurs as you take the photo from the ground.

How far do you need to be away from the Eiffel Tower (height 324 m ) to get its whole length in a photograph?

Answer 1

You would need to be approximately 528 meters away from the Eiffel Tower to capture its entire length in a photograph using a cheap cell phone camera with a single lens.

Step-by-step explanation:

To determine the distance required to capture the entire length of the Eiffel Tower in a photo, we can use the thin lens formula:

`\[ (1)/(f) = (1)/(d_o) + (1)/(d_i) \]`

Given:

`\(d_i = 4.9 \, \text{mm} = 4.9 * 10^(-3) \, \text{m}\) (distance behind the lens)`

`\(h = 324 \, \text{m}\) (height of the Eiffel Tower)`

`\(h_i = 10 \, \text{mm} = 10 * 10^(-3) \, \text{m}\) (height on the sensor)`

First, find the magnification produced by the lens:

`\[ \text{Magnification} = (h_i)/(h) \]`

`\[ \text{Magnification} = (10 * 10^(-3))/(324) = 3.086 * 10^(-5) \]`

Now, calculate the image distance from the lens using the magnification formula:

`\[ \text{Magnification} = (d_i)/(d_o) \]`

`\[ d_o = \frac{d_i}{\text{Magnification}} = (4.9 * 10^(-3))/(3.086 * 10^(-5)) \approx 158.85 \, \text{m} \]`

To capture the entire height of the Tower on the sensor, the object distance should be the sum of the tower's height and the distance behind the lens:

`\[ d_o = d_i + h \]`

`\[ 158.85 \, \text{m} = 4.9 \, \text{mm} + h \]`

`\[ h = 158.85 \, \text{m} - 4.9 \, \text{mm} \approx 158.85 \, \text{m} \]`

Therefore, the distance required to capture the entire length of the Eiffel Tower using the cheap cell phone camera would be approximately
`\(158.85 \, \text{m} + 324 \, \text{m} = 528 \, \text{m}\).`