Use the formula and given information to solve for the indicated variable. Round answers to two decimal places as necessary.
[tex]$I=\frac{P}{4 \pi r^2}$[/tex]
Given: [tex]$P =2,831$[/tex] and [tex]$r =5.4$[/tex].
Calculate I.
Answer (2)
Substitute the given values P = 2831 and r = 5.4 into the formula I = 4 π r 2 P .
Calculate I = 4 × π × ( 5.4 ) 2 2831 .
Evaluate the expression: I ≈ 7.725 .
Round the result to two decimal places: 7.73 .
Explanation
Understanding the Problem We are given the formula I = 4 π r 2 P , and we are asked to calculate I when P = 2831 and r = 5.4 . We will substitute these values into the formula and calculate the result.
Calculations Substitute the given values into the formula: I = 4 × π × ( 5.4 ) 2 2831 Now, we calculate the value of I : I = 4 × π × 29.16 2831 I = 366.435 2831 I ≈ 7.725
Rounding the Result Round the result to two decimal places: I ≈ 7.73
Final Answer Therefore, the value of I is approximately 7.73.
Examples
Understanding the intensity of light or sound as it spreads from a source is a practical application of this formula. For instance, if you have a light bulb (source) emitting a certain amount of light (P), you can calculate how bright the light (I) will be at a certain distance (r) away from the bulb. This is useful in photography, lighting design, and even understanding how radio waves propagate from a transmitter.
To calculate the intensity I using the formula I = 4 π r 2 P , we substitute the values P = 2831 and r = 5.4 . After performing the calculations, we find I ≈ 7.73 .
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