Sahara is a 2005 action-comedy film starring Matthew McConaughey, with the adventurous spirit of the original Indiana Jones movies. Treasure hunters embark on an adventure across the African continent to discover a long lost Confederate gold haul.
Math Scenario 1 – Ironclad Crossing the Atlantic
The movie plot relies on the CSS Texas having made it across the Atlantic ocean and up the Niger river at the end of the US Civil War.
Ironclads were ships that were notoriously likely to sink in rough open ocean waters. But let’s say that the weather for the journey was ideal and the waves were small. How long would it have taken to travel the distance?
Using an online distance calculator, the journey distance by sea is about 10000 km – a long journey even in a modern ship like the one pictured, let alone one from the mid 19th century!
The CSS Texas had a nominal top speed of about 10 knots, but its sister ship the CSS Tennessee II had a top speed of about 5 knots. On a long ocean journey, even ignoring currents, the ship would have been unlikely to maintain a top speed for long periods of time. Let’s say it averages 3.5 knots.
How long does a 10000 km journey take at 3.5 knots?
Journey time = distance / speed
Journey time = 10000 km / 3.5 knots
A knot is equivalent to one nautical mile per hour, or about 1.852 km/hr:
Journey time = 10000 km / (3.5 knots * 1.852 km / nautical mile)
Journey time = 1543 hours
Journey time = 64 days
The two main considerations for such a long journey would have been how much coal it could have carried to power the steam engines, and how much food the crew would require. Given ironclads were not meant to travel on the open ocean, such a journey would have been very perilous indeed.
Math Scenario 2 – Vaporizing Toxic Waste with Solar Power
The baddies in Sahara are planning to vaporise toxins using their massive solar farm, something like the one in the picture above. To vaporize the toxic barrels, which are put in an incinerator at the top of the tower, the reflecting solar panels need to be angled just right. We can calculate the angle of elevation a for the two solar panels in the image below:
For the panel on the left, we can work out the angles in the triangle using trigonometry.
We know the opposite and adjacent sides of the triangle for the angle (90 – a), so we can use tan (SOHCAHTOA) to calculate what the angle 90-a is:
tan(90 – a) = 250 / 200
Take the arctan of both sides of the equation:
90 – a = atan(250 / 200)
a = 90 – atan(250 / 200)
a = 41.40 degrees
The panel on the left should be elevated 41.4 degrees above the horizontal to target the top of the tower.
The panel on the right is further from the tower, so the numbers change, but the calculation is otherwise identical:
tan(90 – a) = 250 / 300
90 – a = atan(250 / 300)
a = 90 – atan(250 / 300)
a = 50.91 degrees
Since it’s further from the tower, the panel has to be angled about 9 degrees more steeply than the closer panel.
Math Scenario 3 – Toxic Spread into the Atlantic
At one point in the movie it becomes apparent that the toxic waste could spread out of the river and catastrophically throughout the Atlantic ocean, killing all marine life and causing international chaos.
Let’s say the analysts work out that the area of affected ocean follows an exponential growth curve:
A = 1 * 3^t
A is the area of affected ocean in square kilometres, and t is the time in days from when the toxic spill first hits the ocean at the mouth of the river
The area of the Atlantic Ocean is approximately 106500000 square kilometres. We can work out how long it will take for the spill to spread throughout the entire Atlantic Ocean:
Firstly, using trial and error:
10 days: A = 1 * 3^10 = 59049 km^2
15 days: A = 1 * 3^15 = 14348907km^2
18 days: A = 1 * 3^18 = 387420489 km^2
17 days: A = 1 * 3^17 = 129140163 km^2
16 days: A = 1 * 3^16 = 43046721 km^2
So sometime between the 16th and 17th day the entire Atlantic Ocean will be affected.
We can also solve this problem using logarithms:
A = 1 * 3^t
log(A) = log(1 * 3^t)
log(A) = t * log(1 * 3)
t = log(A) / log(3)
t = log(106500000) / log(3)
t = 16.82 days
Real Life Example – Algal Growth
Algae (the green stuff you sometimes see on ponds) can grow exponentially under certain circumstances, described by this growth formula:
N = S * e^(kt)
where N is the population at time t, S is the starting population, k is some constant value and t is the time that has past
However, exponential growth in real natural systems usually only occurs for quite short periods of time. Typically, a lack of food or space, or increases in the number of predators means that the growth slows to below exponential growth or even reverses.