An island is 3 miles from the nearest point p on a straight shoreline. A town is 10 miles down the shore from p.
a) Enter a function T(x) that describes the total amount of time the trip takes as a function of the distance x.
b) What is the domain of T?
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a) To find the function T(x), we need to consider the two parts of the trip: the boat ride from the island to point p on the shore, and the car ride from point p to the town.
The boat ride is a straight line from the island to a point on the shore, which is a distance of sqrt(x^2 + 3^2) miles (using the Pythagorean theorem). If we let the speed of the boat be b miles per hour, then the time for the boat ride is sqrt(x^2 + 3^2)/b hours.
The car ride is a straight line from point p to the town, which is a distance of 10 - x miles. If we let the speed of the car be c miles per hour, then the time for the car ride is (10 - x)/c hours.
So the total time for the trip is T(x) = sqrt(x^2 + 3^2)/b + (10 - x)/c.
b) The domain of T is the set of all possible values of x. Since x represents the distance along the shore from point p to the point where the boat lands, it must be between 0 and 10 miles (inclusive). So the domain of T is [0, 10].
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