The provided equation models the length of the scalloped hammerhead shark in relation to its age. Here, the length of the shark, represented as 'l', is 175 centimeters. The formula is written as l=266-219*e^(-0.05*t), and we want to solve for t, or the age of the shark.
Step 1:
First, we plug in the given length of the shark into the equation. This changes the formula to:
266 - 219*e^(-0.05*t) = 175
Step 2:
Rearrange this to find the exponential expression on one side of the equal sign:
219 * e^(-0.05*t) = 266 - 175
Step 3:
Divide both sides by 219 to isolate the exponential part of the formula:
e^(-0.05*t) = (266 - 175) / 219
Step 4:
Next, we need to extract t from the exponent. To do this, we take the natural logarithm of both sides. The natural logarithm of e^x is x, so this will pull our t out of the exponent on the left-hand side:
-0.05*t = ln((266 - 175) / 219)
Step 5:
Finally, to solve for t, you divide both sides by -0.05:
t = ln((266 - 175) / 219) / -0.05
When we calculate the result, we find that the shark is approximately 17.56 years old. Therefore, based on this mathematical model, a scalloped hammerhead shark that is 175 centimeters long is approximately 17.56 years old.