Question 6. The initial dryweight of Botryococcus braunii, a green microalgea, is 50 mg/L. Under the right conditions of light, temperature, salinity and pH, the algea triples every three hours for the first 15 hours of growth. (3 marks each) a. Determine the rule domain, and range of the function model b. Graph the function model

The initial concentration is 50mg/L, and underideal conditions, the concentration triples every three hours forthe first 15 hours. So, after 3 hours, the concentration will be50x3 = 150mg/LNow, let W be the weight and tbe the number of hours. Let, the required function model beW(t)=50 e^{k t}Then clearly, when t = 0, W = 50. Now, whent = 3, W = 150, soW(3)=150 \Rightarrow 50 e^{3 k}=150 \Rightarrow e^{3 k}=3 \Rightarrow k=\frac{\ln 3}{3}=0.366Thus, the function rule is given byW(t)=50 e^{0.366 t}Now, the given rate of tripling every three hours is valid onlyfor the first 15 hours, so the domain of thefunction is given byDomain W=[0,15]Now, clearly, W is an exponentially increasingfunction, and at t = 0, W = 50. So to find therange, we need to find the W when t =15, soW(15)=50 e^{0.366 \times 15}=50 \times 243=12150 \mathrm{mg} / \mathrm{L}So, range of W is given byRange W=[50,12150this completes part (a)Now, graphing the function we get ...