The Deepwater Horizon oil spill began on April 20, 2010, and the well was not declared sealed until September of 2010. Suppose that the spill was relatively circular in shape, and that the radius of the spill increased at a rate of approximately 7 miles per day. a) Fill in the table below showing the radius in miles, $r$, of the oil spill $t$ days after April 20. b) Write a formula for $r$ as a function of $t$. \[ r(t)= \] c) Venice, Louisiana is approximately 50 miles from the site of the well. Approximately how long did it take for oil to wash up on the shores of Venice? Please round your answer up to the next whole number. days d) Assuming the spill was circular, write a formula for the area, $A$, of the oil spill as a function of $t$. Hint: recall that the area of a circle is given by $\pi r^{2}$. \[ A(t)= \] e) How many full days after the initial spill would it take to guarantee the area of the spill was 6000 square miles? Please round your answer up to the next whole number. days
Solution
Step 1 ：Create a table for the first 10 days after the spill. The table will have two columns: 'Day' and 'Radius'. The 'Day' column will contain the number of days from April 20, and the 'Radius' column will contain the radius of the oil spill on that day. The radius can be calculated by multiplying the number of days by 7.
Step 2 ：The table is as follows: \[\begin{array}{|c|c|} \hline \text{Day} & \text{Radius} \\ \hline 1 & 7 \\ 2 & 14 \\ 3 & 21 \\ 4 & 28 \\ 5 & 35 \\ 6 & 42 \\ 7 & 49 \\ 8 & 56 \\ 9 & 63 \\ 10 & 70 \\ \hline \end{array}\]
Step 3 ：Write a formula for the radius, $r$, as a function of time, $t$. Since the radius increases at a rate of 7 miles per day, the formula is $r(t) = 7t$.
Step 4 ：Venice, Louisiana is approximately 50 miles from the site of the well. To find out how long it took for oil to wash up on the shores of Venice, solve the equation $r(t) = 50$ for $t$. This gives $t = \frac{50}{7} \approx 7.14$. Since we need to round up to the next whole number, it took approximately 8 days for oil to wash up on the shores of Venice.
Step 5 ：Assuming the spill was circular, the area, $A$, of the oil spill as a function of $t$ can be found using the formula for the area of a circle, $A = \pi r^{2}$. Substituting $r(t) = 7t$ into this formula gives $A(t) = \pi (7t)^{2} = 49\pi t^{2}$.
Step 6 ：To find out how many full days after the initial spill it would take to guarantee the area of the spill was 6000 square miles, solve the equation $A(t) = 6000$ for $t$. This gives $t = \sqrt{\frac{6000}{49\pi}} \approx 13.92$. Since we need to round up to the next whole number, it would take approximately 14 days for the area of the spill to reach 6000 square miles.
