A person needs to swim to shore from Point A and walk to Point B. They can swim at 2mph and walk at 4 mph. Write an expression for the total time their journey will time their journey will take in terms of x. Then use your graphing calculator to estimate the value of x which will make the trip last a minimum amount of time. Also give the total minimum time for the trip.
I assume the person swims from A to C, which is a distance x from the point D on shore closest to A, then walks from C to B.
Note the lamentable lack of any actual numbers, which limits my ability to provide a numeric answer.
Without loss of generality, let AD=1, and CB = ax for some constant a.
You can scale the diagram as needed
distance from A to C: √(1+x^2)
distance from C to B: ax
time t for the trip
t = 1/2 √(1+x^2) + 1/4 ax
dt/dx = x/[2√(1+x^2)] + a/4
= [2x + a√(1+x^2)] / 4√(1+x^2)
dt/dx = 0 when x = a/(4-a^2)
To find the expression for the total time of the journey in terms of x, we need to consider the distances for both swimming and walking.
Let the distance from Point A to shore be d1, and the distance from shore to Point B be d2.
The time taken to swim from Point A to shore at a speed of 2 mph can be calculated as t1 = d1 / 2.
The time taken to walk from shore to Point B at a speed of 4 mph can be calculated as t2 = d2 / 4.
The total time for the journey, T, can be expressed as the sum of the swimming time and walking time:
T = t1 + t2 = d1/2 + d2/4
To estimate the value of x that minimizes the total time, you can graph the function T(x) = d1(x)/2 + d2(x)/4 on a graphing calculator. This requires having the specific values for d1(x) and d2(x) as functions of x.
You can then use the minimum function on your graphing calculator to find the x-value that minimizes T(x). This will give you the value of x that minimizes the total time of the journey.
Once you have the x-value, substitute it into the expression for T(x) to get the total minimum time for the trip.