For a square, if the area is $49$, then the side length is $\sqrt{49}=7$ meters. The perimeter of a square is $4$ times the side length, so the perimeter is $4(7)=28$ meters.

Set the model equal to 40: $5(2^t)=40$. Dividing both sides by 5 gives $2^t=8$. Since $2^3=8$, it follows that $t=3$. So the substance reaches 40 grams after $3$ hours.

To undo the square root, square both sides of the equation:

This gives $x=81$.

Since $(a,8)$ is on the graph of $y=2^x$, we have $2^a=8$. Because $8=2^3$, it follows that $a=3$. Since $(16,b)$ is on the graph of $y=\sqrt{x}$, we have $b=\sqrt{16}=4$. Therefore, $a+b=3+4=7$.

Substitute $y=2^x$ into $x+y=6$ to get

Now test values of $x$ that make the exponential expression easy to evaluate. If $x=2$, then

so $x=2$ works. Then

Since the system is stated to have exactly one solution, this is the only possible value of $y$. Therefore, the correct answer is $4$.