$\mathrm{Let}{\mathrm{ P}}_k\text{ represent the perimeter of a square with sides of length}k.\text{The value of the expression}$
P_1+P_2+P_3+\cdots +P_{10}  $\text{is --------}$

The correct option is 220.

Let us break down the problem step-by-step to understand how to arrive at this answer.

First, we are given that $P_k$ represents the perimeter of a square with sides of length $k$.

Recall that a square has four equal sides. The perimeter of a square with a side length of $k$ is the sum of the lengths of all its four sides. Therefore, the formula for the perimeter is:
$$P_k=4k$$

We need to find the value of the sum:
$$S=P_1+P_2+P_3+\cdots +P_{10}$$

Substituting the perimeter formula $P_k=4k$ into the expression, we get:
$$S=(4\times 1)+(4\times 2)+(4\times 3)+\cdots +(4\times 10)$$

We can factor out the common constant multiplier, which is 4:
$$S=4\times (1+2+3+\cdots +10)$$

Now, we need to calculate the sum of the first 10 positive integers: $1+2+3+\cdots +10$.

Using the formula for the sum of the first $n$ natural numbers, which is:
$$\sum _{i=1}^{n}i=\frac{n(n+1)}{2}$$

For $n=10$, this calculation yields:
$$1+2+3+\cdots +10=\frac{10\times (10+1)}{2}=\frac{10\times 11}{2}=5\times 11=55$$

Substitute this sum back into our equation for $S$:
$$S=4\times 55$$

Multiplying the terms, we get:
$$S=220$$

Thus, the final value of the expression is 220.