## To find the length of the string in question 1, we can use the equation:

v = f * Î»

where:

v is the speed of the wave,

f is the frequency of the wave, and

Î» is the wavelength of the wave.

In this case, the frequency is given as 260 Hz and the wavelength is given as 0.60 m.

To find v, we can use the equation:

v = âˆš(T / Î¼)

where:

T is the tension in the string, and

Î¼ is the mass per unit length of the string.

In this case, the tension is given as 180 N.

To find Î¼, we can rearrange the equation:

Î¼ = T / v^2

Now let's substitute the given values into the equations to find the length of the string:

1. Calculate v:

v = âˆš(T / Î¼) = âˆš(180 / Î¼)

2. Calculate Î¼:

Î¼ = T / v^2 = 180 / v^2

3. Substitute v into the equation for Î¼:

Î¼ = 180 / (âˆš(180 / Î¼))^2 = 180 / (180 / Î¼) = Î¼

Now we have an equation:

Î¼ = Î¼

Since the Î¼ terms on both sides cancel out, we can see that the mass per unit length of the string is equal to itself.

Therefore, there is no given information about the mass per unit length of the string, so we cannot determine the length of the string without this information.

Moving on to question 2:

To find the sound intensity level at a distance of 85 m from the engine, we can use the inverse square law. This law states that the sound intensity decreases with the square of the distance.

The equation for the sound intensity level (L) relative to the threshold of hearing is:

L = 10 * log10(I / I0)

where:

I is the sound intensity in watts per square meter, and

I0 is the reference intensity, which is the threshold of hearing (typically 1 x 10^-12 W/m^2).

In this case, the sound energy emitted by the engine is given in joules per second. To find the sound intensity (I), we can divide the sound energy by the surface area of a sphere with a radius equal to the distance from the engine.

The formula for the surface area of a sphere is:

A = 4Ï€r^2

where:

A is the surface area of the sphere, and

r is the radius of the sphere.

Let's start by finding the sound intensity (I):

1. Calculate the surface area of the sphere:

A = 4Ï€r^2 = 4 * 3.14159 * (85)^2

2. Divide the sound energy emitted per second by the surface area of the sphere:

I = sound energy / A

3. Substitute the given value of the sound energy and the calculated value of the surface area into the equation:

I = sound energy / A

4. Calculate the sound intensity level using the equation:

L = 10 * log10(I / I0)

Now, let's plug in the numbers and calculate the sound intensity level at a distance of 85 m from the engine.