{Use of Tech } Min​​imizing sound intensity Two sound speakers are 100 m apart and one speaker is three times as loud as the other speaker. At what point on a line segment between the speakers is the sound intensity the weakest? (Hint: Sound intensity is directly proportional to the sound level and inversely proportional to the square of the distance from the sound source.)

Verified step by step guidance

1. Define the variables: Let the intensity of the quieter speaker be I and the louder speaker be 3I, where I is the intensity of the quieter speaker.
2. Set up the distance: Let x be the distance from the quieter speaker to the point where the sound intensity is being measured. Therefore, the distance from the louder speaker to this point is (100 - x) meters.
3. Write the expression for sound intensity: The total sound intensity at point x can be expressed as I_total = I / x^2 + 3I / (100 - x)^2.
4. To find the minimum intensity, take the derivative of I_total with respect to x and set it equal to zero to find critical points: d(I_total)/dx = 0.
5. Analyze the critical points and endpoints to determine where the sound intensity is minimized, ensuring to check the second derivative or use the first derivative test for confirmation.