Can Anyone get me started on this problem?

[sarahjohnson]

It came with this diagram:

I'm not sure even where to get started with finding the equation...like how to relate her rowing and walking. Any hints would be appreciated thank you.

 

[MarkFL]

What kind of triangle is \(\displaystyle \Delta ABC\)?

 

[DrPhil]

Brooke is standing at the location "A" on the shore of Circle Lake. The lake has a circular shoreline and a radius of 2 miles. In this problem, we will study the minimum and maximum time it takes Brooke to go from the location "A" to the location "C" diametrically opposite, subject to several scenarios regarding how fast she can walk and row. In all cases, assume Brooke rows in a straight line and walks along a portion of the circumference, as indicated by the arrows. We have labeled an angle q in the picture. Note:
(i) When q=0, Brooke rows straight from "A" to "C" and never walks.
(ii) When q=pi/2, Brooke walks from "A" to "C" along the upper half circumference and never rows.
(iii) When q is strictly between 0 and pi/2, Brooke rows from "A" to "B" and then walks from "B" to "C"

In what follows, be careful to convert "ft/sec" to "miles/hr" units using the fact that there are 5280 feet/mile and 3600 seconds/hour.(a) Assume that Brooke can walk c ft/sec and row d miles/hr. Find a formula for the time (in hours) it takes Brooke to go from "A" to "C" via "B"; your formula should involve c,d and q:

It came with this diagram:

I'm not sure even where to get started with finding the equation...like how to relate her rowing and walking. Any hints would be appreciated thank you.

What kind of triangle is [FONT=MathJax_Main]Δ[/FONT][FONT=MathJax_Math]A[/FONT][FONT=MathJax_Math]B[/FONT][FONT=MathJax_Math]C[/FONT]?​

Waiting to see some work...

Wasn't enough clue yet??

As a function of the angle q, what is the length of the triangle leg AB? Rowing time = ...

How is the arc length BC measured from the center of the circle related to angle q? Walking time = ...
​

 

[sarahjohnson]

Sorry I was at school the entire day.
So it is a right triangle!
Then tan(q)=BC/AB? and AB=BC/Tan(q) ?

and then AB=d which the rowing time? I don't think that's right...but I should replace AB and BC with something right?

Also didn't learn about how to do arc length but I will try to google it and figure it out.

I'm sorry I'm slow at word problems

 

[edumat]

The first step is to calculate the distance to row:

As you can se above, \(\displaystyle cos(q) = x/2 => x = 2 cos(q) => d_{row} = 2x => d_{row} = 4 cos(q) (miles)\)

Now we calculate the distance to walk... if we put the angle in the center of the circunference, that would be 2q, as you can see below

As you can see, x represents a part of the circunference. The circunference is \(\displaystyle 2πr = 4π (miles)\)
The full circunference would be \(\displaystyle 2π(rad)\). So \(\displaystyle 2q/2π (rad) = q/π (rad)\) is the fraction of the circunference we're interested in.
Now we have: \(\displaystyle d_{walk} = x = \frac{q}{π}4π = 4q (miles)\)

Since we have both distances, the next step is to work with the speeds.
d (row) is already in miles/hour.
c (walk) is in ft/sec, so we convert:
\(\displaystyle c \frac{ft}{sec}\frac{1 mile}{5280 ft}\frac{3600 sec}{1 hour} = \frac{15}{22}c (\frac{mile}{hour})\)

We know that time is distance/speed, so, putting it all together, we have the equation of time:
\(\displaystyle T(q) = \frac{4cos(q)}{d}+\frac{4q}{(15/22) c} = \frac{4cos(q)}{d}+\frac{88q}{15c} (miles)\)

 

[sarahjohnson]

Thank you for that! that makes sense now on how to relate them. It turned out easier than I thought
Also thanks to everyone who was helping me.