Math Is Fun Forum

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Build An Algebraic Model

An Island is 2 miles from the nearest point P on a straight shoreline. A town is 12 miles down the shore from P.

A. If a person can row a boat at an average speed of 3 mph and the same person can walk 5 mph, build a model that expresses the time T that it takes to go from the Island to town as a function of the distance x from P to where the person lands the boat.

B. Find the domain of T.

Let me see.

Let x = the landing point between P and the nearest town, where x is
0 <= x <= 12.

Let A = path of boat = sqrt{4 + x^2}.

Let B = path of walking = 12 - x

Let T(x) = total time

T(x) = A/(boat mph) + B/(walking mph)

T(x) = sqrt{4 + x^2}/3 + (12 - x)/5

Is this right?

Part B

I say domain = 0 miles to 12 miles.

So, domain = [0, 12].

You say?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,637

Re: Build An Algebraic Model

Part A is correct. That was tough so well done!

The domain is for time not distance.  What is T when x=0 and what is T when x = 12. That gives you the limits.

I've just graphed it and it doesn't give the full domain. The graph goes down to a local minimum, then rises again. So what I said misses that lowest point.  You'll have to find the T value there, either by using the graph or calculus.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Build An Algebraic Model

Part A is correct. That was tough so well done!

The domain is for time not distance.  What is T when x=0 and what is T when x = 12. That gives you the limits.

I've just graphed it and it doesn't give the full domain. The graph goes down to a local minimum, then rises again. So what I said misses that lowest point.  You'll have to find the T value there, either by using the graph or calculus.

Bob

Ok. I will graph it using Desmos and see what information the picture gives.