A population of bacteria is initially 3000 . After two hours the population is 1500 . If this rate of decay continues, determine the exponential function of the form $f(t)=a(b)^{c t}$ that represents the size of the bacteria population after $t$ hours.

Step 1 ：Given that the population of bacteria is initially 3000 and after two hours the population is 1500, we can use these two points to determine the exponential function of the form \(f(t)=a(b)^{c t}\).

Step 2 ：We know that at \(t=0\), \(f(t)=3000\) and at \(t=2\), \(f(t)=1500\).

Step 3 ：Substituting these values into the function, we get two equations: \(3000=a(b)^{c*0}\) and \(1500=a(b)^{c*2}\).

Step 4 ：The first equation simplifies to \(3000=a\) because any number raised to the power of 0 is 1.

Step 5 ：Substituting \(a=3000\) into the second equation, we get \(1500=3000(b)^{2c}\).

Step 6 ：Solving the second equation for \(b^{2c}\), we get \(b^{2c}=\frac{1500}{3000}=\frac{1}{2}\).

Step 7 ：Taking the square root of both sides, we get \(b^c=\sqrt{\frac{1}{2}}\).

Step 8 ：Since we want the function in the form \(f(t)=a(b)^{c t}\), we need to find the value of \(b\) when \(c=1\). So, \(b=\sqrt{\frac{1}{2}}\).

Step 9 ：Therefore, the exponential function that represents the size of the bacteria population after \(t\) hours is \(f(t)=3000(\sqrt{\frac{1}{2}})^t\).

Step 10 ：At \(t=0\), \(f(t)=3000(\sqrt{\frac{1}{2}})^0=3000\), which is the initial population of bacteria.

Step 11 ：At \(t=2\), \(f(t)=3000(\sqrt{\frac{1}{2}})^2=3000*\frac{1}{2}=1500\), which is the population of bacteria after two hours.

Step 12 ：\(\boxed{f(t)=3000(\sqrt{\frac{1}{2}})^t}\) meets the requirements of the problem.