Question

a speaker is producing spherically symmetric sounds with a total power of 61.5 w. the speaker is located at the centre of a sphere with a radius of 1.20 m. if a square detector at the edge of the sphere measures a power of 4.55 w, what is the total area of this detector?give your answer in metres squared to 2 decimal places. do not include units in your answer.

Answer 1

The total area of a square detector measuring a power of 4.55 W at the edge of a sphere, with the speaker at the center emitting 61.5 W of power, is approximately 0.43 m² when the radius of the sphere is 1.20 m.

Step-by-step explanation:

To find the total area of the square detector that measures a power of 4.55 W, we begin by understanding that the power from the speaker distributes uniformly across the surface area of the sphere. Since the speaker's power at the center of the sphere is given to be 61.5 W, we use the formula for the surface area of a sphere A = 4πr², where r is the radius of the sphere.

The radius of the sphere is 1.20 m, so the total surface area A is:
A = 4π(1.20)² = 4π(1.44) = 18.096π m².

Now we can calculate the intensity of sound at the surface of the sphere, which is the power spread over the area of the sphere:
I = P/A,
where P = 61.5 W and A = 18.096π m².

The intensity is:
I = 61.5 W / (18.096π m²) = 61.5 / (18.096π) W/m².

With the intensity known, we can calculate the area of the detector by rearranging the intensity formula to solve for A:
A = P/I,
where P = 4.55 W is the power detected and I is the intensity we just calculated.

By substituting the values we have, we get:
A (area of the detector) = 4.55 W / (61.5 / (18.096π) W/m²) = (4.55 * 18.096π) / 61.5 m²,
which calculates to an area of approximately 0.43 m² to two decimal places.