**Solution:**

To calculate the length of the path, we need to find the total distance covered by the body during one complete cycle of SHM.

**Understanding Simple Harmonic Motion (SHM):**

Simple Harmonic Motion is a type of periodic motion where the restoring force is directly proportional to the displacement and acts towards the mean position. It can be represented by the equation:

*a = -ω²x*

Where:
*a* is the acceleration,
*ω* is the angular frequency, and
*x* is the displacement from the mean position.

**Finding the Angular Frequency (ω):**

Given that the body has a velocity of 10 cm/s when the displacement is 3 cm, we can use the formula for velocity in SHM to find the angular frequency:

*v = ω√(A² - x²)*

Where:
*v* is the velocity,
*ω* is the angular frequency,
*A* is the amplitude, and
*x* is the displacement.

Plugging in the values, we have:

10 = ω√(A² - 3²)

Similarly, when the body has a velocity of 7 cm/s at a displacement of 4 cm, we can use the same formula to find ω:

7 = ω√(A² - 4²)

We now have two equations with two unknowns, ω and A. Let's solve these equations simultaneously to find the values.

**Solving the Equations:**

From the first equation, we can square both sides to eliminate the square root:

100 = ω²(A² - 3²)
100 = ω²(A² - 9)
100 = ω²A² - 9ω²

From the second equation, we can square both sides again:

49 = ω²(A² - 4²)
49 = ω²(A² - 16)
49 = ω²A² - 16ω²

Now, we can subtract the second equation from the first equation to eliminate ω²A²:

51 = 7ω²
ω² = 51/7
ω = √(51/7)

Substituting this value back into either of the original equations, we can solve for A:

10 = ω√(A² - 3²)
10 = √(51/7)√(A² - 9)
10 = √(51(A² - 9)/7)

Squaring both sides again:

100 = 51(A² - 9)/7
700 = 51(A² - 9)
A² - 9 = 700/51
A² - 9 = 13.73
A² = 13.73 + 9
A² = 22.73
A = √22.73

**Calculating the Length of the Path:**

Now that we have the amplitude (A) and angular frequency (ω), we can calculate the length of the path.

The length of the path covered by the body during one complete cycle of SHM can be given as:

Length = 4A

Substituting the value of A, we have:

Length = 4√22.73

Hence, the length of the path covered by the body is 4 times the square root of 22.73.