Because a stadium will be used at night, lights have been installed. The brightness (B) of a light varies directly with the intensity (I) of the light and inversely as the square of the distance (D) from the light. Brightness (B) is measured in light energy per unit and intensity (I) is measured in candle power. At a distance of 40 feet, a light meter gives a reading if 12 light energy units for a 160-candle power lamp
Part A: Write an equation that can be used to determine the constant of variation (k), then solve it for k.
Part B: Find the reading of a 90 candle power lamp at the distance of 60 feet.
B = something * I
B = something / D^2
so
B = k I / D^2
the question is what is k?
12 = k *160 / (40*40)
so k = 12 * 40 * 40 / 160
k = 120 in these units
B = 120 * 90 / (60*60) = 2 * 3 / 2 = 3
Part A:
To write the equation, we know that the brightness (B) varies directly with the intensity (I) and inversely as the square of the distance (D). This can be written as:
B = k * (I / D^2)
where k is the constant of variation.
Given that at a distance of 40 feet, a light meter gives a reading of 12 light energy units for a 160-candle power lamp, we can plug in these values into the equation:
12 = k * (160 / 40^2)
Simplifying:
12 = k * (160 / 1600)
12 = k / 10
Multiplying both sides by 10:
120 = k
So, the constant of variation (k) is 120.
Part B:
To find the reading of a 90 candle power lamp at the distance of 60 feet, we can use the same equation:
B = k * (I / D^2)
Plugging in the values:
B = 120 * (90 / 60^2)
Simplifying:
B = 120 * (90 / 3600)
B = 120 * 0.025
B = 3
Therefore, the reading of a 90 candle power lamp at a distance of 60 feet is 3 light energy units.
Part A:
Let's denote the constant of variation as k. We know that brightness (B) varies directly with intensity (I) and inversely as the square of the distance (D).
This can be expressed as:
B = k * (I / D^2)
Using the given values at a distance of 40 feet (D) and an intensity of 160 candle power (I), we can substitute these values into the equation:
12 = k * (160 / 40^2)
Simplifying further:
12 = k * (160 / 1600)
12 = k * 0.1
To solve for k, divide both sides of the equation by 0.1:
k = 12 / 0.1 = 120
Therefore, the constant of variation is k = 120.
Part B:
Now that we have determined the constant of variation (k), we can use it to find the reading of a 90 candle power lamp at the distance of 60 feet.
Using the equation B = k * (I / D^2), we substitute the values:
B = 120 * (90 / 60^2)
B = 120 * (90 / 3600)
Simplifying further:
B = 120 * 0.025
B = 3
Hence, the reading of a 90 candle power lamp at a distance of 60 feet would be 3 light energy units.