Task B4.2  Since the iceberg is melting over time during the journey, the next level of complexity is to include the time-dependence of the iceberg’s speed if the tug pulls at constant power.  (a) Write down the iceberg radius as a function of time, and then the pulling speed as a function of time.  (b) How are the total journey time, journey length and the time-dependent speed related? Derive an expression for the final iceberg radius as a function of initial iceberg radius. (Hint: the final radius is related to initial radius and the journey time).  (c) Now you know how the final radius depends on initial radius, you can calculate the income from selling the delivered iceberg and the fuel costs of the tug. Develop a Matlab script to plot the gross profit (sales minus fuel costs) as a function of initial iceberg size.  (d) Use your Matlab work to find the following: i. The minimum initial iceberg size for the sales revenue to be greater than the given fuel costs (positive gross profit); ii. The initial iceberg radius and associated gross profit if the total journey time is limited to 30 days due to seasonal factors; iii. What these will be if the fuel price increases to 140p per litre.

## Heading 2<br /><br />(a) To write down the iceberg radius as a function of time, we need to consider the melting rate. Let's assume that the iceberg radius decreases linearly with time. If the initial radius is denoted as R0 and the melting rate as m, then the iceberg radius as a function of time t can be expressed as:<br /><br />R(t) = R0 - mt<br /><br />The pulling speed as a function of time can be derived from the power equation. Assuming the tug pulls at constant power P, the pulling speed v can be expressed as:<br /><br />v(t) = P / (2πR(t))<br /><br />(b) The total journey time, journey length, and time-dependent speed are related as follows: The journey length is the integral of the pulling speed over time. The total journey time is the journey length divided by the time-dependent speed. Mathematically, we have:<br /><br />Journey length = ∫[0,T] v(t) dt<br /><br />Total journey time = Journey length / v(T)<br /><br />To derive an expression for the final iceberg radius as a function of the initial iceberg radius, we need to consider the relationship between the journey time and the final radius. Let's assume that the journey time is denoted as T and the final radius as Rf. Then, we can express the final radius as:<br /><br />Rf = R0 - mT<br /><br />(c) To calculate the gross profit as a function of the initial iceberg size, we need to consider the income from selling the delivered iceberg and the fuel costs of the tug. Let's assume that the income per unit volume of the delivered iceberg is I and the fuel cost per unit distance is C. The gross profit GP can be expressed as:<br /><br />GP = I * (4/3)πRf^3 - C * Journey length<br /><br />You can develop a MATLAB script to plot the gross profit as a function of the initial iceberg size by varying the initial radius R0 and calculating the corresponding final radius Rf using the expression derived in part (b). Then, substitute the values into the gross profit equation and plot it.<br /><br />(d) Using your MATLAB work, you can find the following:<br /><br />i. The minimum initial iceberg size for the sales revenue to be greater than the given fuel costs (positive gross profit) can be determined by finding the initial radius R0 that makes the gross profit GP greater than zero.<br /><br />ii. To find the initial iceberg radius and associated gross profit if the total journey time is limited to 30 days, you can set the total journey time T to 30 and calculate the corresponding initial radius R0 and gross profit GP using the derived expressions.<br /><br />iii. To determine the values if the fuel price increases to 140p per litre, you can update the fuel cost per unit distance C in the gross profit equation and recalculate the gross profit for different initial iceberg sizes.<br /><br />By analyzing the results, you can make informed decisions regarding the minimum initial iceberg size, the impact of journey time limitations, and the effect of fuel price changes on gross profit.