Answer: \(A=1000\), \(b=100\), so \(y=1000\times100^{\sqrt{x}}\).

Transform the given relationship into a straight-line form. The gradient and intercept of the straight line then give the constants in the original model.

Since \(\lg y\) is plotted against \(\sqrt{x}\), write the straight line as

\(\lg y=m\sqrt{x}+c.\)

The line passes through \((1,5)\) and \((2.5,8)\), so

\(m=\frac{8-5}{2.5-1}=2.\)

Using \((1,5)\),

\(5=2(1)+c,\)

so \(c=3\). Hence

\(\lg y=2\sqrt{x}+3.\)

Therefore

\(y=10^{2\sqrt{x}+3}=10^3\left(10^2\right)^{\sqrt{x}}.\)

So

\(y=1000\times100^{\sqrt{x}}.\)

Thus \(A=1000\) and \(b=100\).