The aperture setting, or f-stop, of a digital camera controls the amount of light exposure on the sensor. Each higher number of the f-stop doubles the amount of light exposure. The formula n=log_2(1/p) represents the change in the number, n, of the f-stop needed, where p is the amount of light exposed on the sensor.
a) a photographer wishes to change the f-stop to accomodate a cloudy day in which only 1/4 of the sunlight is available. How many f-stops does the setting need to be moved?
My answer:
2 (which is correct)
b) if the photographer decreases the f-stop by four settings, what fraction of the light is allowed to fall on the sensor?
I need help on part b. My attempt is this:
-4p=log_2(1/p)
then I solved for p, and got -1/2 but it is wrong.
4=log(base 2) 1/p
2^4 = 1/p
16 = 1/p
16p = 1
p = 1/16
To solve part b, we can start by rearranging the formula to isolate "p":
n = log_2(1/p)
We want to decrease the f-stop by four settings, which means we are looking for the value of "p" when "n" is equal to -4.
-4 = log_2(1/p)
Next, we can rewrite the equation using the exponential form of logarithms:
2^(-4) = 1/p
Simplifying further:
1/16 = 1/p
Cross multiplying:
p = 16
Therefore, when the f-stop is decreased by four settings, the fraction of light allowed to fall on the sensor is 1/16.
To solve part b) of the question, we need to understand that when the f-stop is decreased by four settings, the amount of light exposure on the sensor decreases. We can use the formula n=log_2(1/p), as provided.
Let's proceed step by step:
Step 1: Start with the given formula: n = log_2(1/p).
Step 2: Substitute the new value for n, which is -4, since the f-stop is decreased by four settings: -4 = log_2(1/p).
Step 3: Rewrite the equation using exponential form: 2^(-4) = 1/p.
Step 4: Simplify the equation: 1/2^4 = 1/p. Since 2^4 = 16, we have 1/16 = 1/p.
Step 5: Cross-multiply: p = 16.
So, when the photographer decreases the f-stop by four settings, 1/16th of the light is allowed to fall on the sensor.
Therefore, the correct answer is that the fraction of light allowed to fall on the sensor is 1/16, not -1/2.