Step 1 :The population of a colony of mosquitoes obeys the law of uninhibited growth, which means that the population grows exponentially over time. The general form of an exponential growth function is given by: \(N(t) = N0 * e^{kt}\), where \(N(t)\) is the population at time \(t\), \(N0\) is the initial population (at time \(t = 0\)), \(k\) is the growth rate, and \(t\) is the time.
Step 2 :We are given that the initial population (\(N0\)) is 1000 mosquitoes and that the population after 1 day (\(N(1)\)) is 1400 mosquitoes. We can use these values to find the growth rate (\(k\)).
Step 3 :First, we substitute \(N0 = 1000\) and \(N(1) = 1400\) into the exponential growth function: \(1400 = 1000 * e^{k*1}\).
Step 4 :Next, we solve for \(k\): \(e^k = 1400 / 1000 = 1.4\). Taking the natural logarithm of both sides gives: \(k = \ln(1.4) \approx 0.3365\).
Step 5 :Now that we have the growth rate, we can find the population after 4 days (\(N(4)\)) by substituting \(N0 = 1000\), \(k = 0.3365\), and \(t = 4\) into the exponential growth function: \(N(4) = 1000 * e^{0.3365*4}\).
Step 6 :Calculating this gives: \(N(4) \approx 1000 * e^{1.346} \approx 3842.8\). Rounding to the nearest whole number gives approximately 3843 mosquitoes.
Step 7 :So, the population of the mosquitoes after 4 days will be \(\boxed{3843}\) mosquitoes.