Tricky(?) Crocodile Maths Exam Question

[Gerald H]

Better solved with a calculator or pencil & paper?

http://www.telegraph.co.uk/education/edu...tears.html

[peacecalc]

Hello all,

it is another version of the problem, which I call the "lifeguard" problem. In reality the lifeguard takes the shortest way through the water (and the longest way over the beach), because that is faster as he tries to find the best (fastest) way with mathematics (the drowning man/woman will be gone to water).

It is the mechanical analogy of snell's law in optics, both have the same solution.

[Gerald H]

Concerning lifeguards you are certainly right, peacecalc, but concerning crocodiles remember they have had many millions of years of evolution to sharpen their skills at calculating minima, and as it's their own survival involved in the success of their reckoning they are very proficient.

[CR Haeger]

(10-09-2015 03:47 PM)Gerald H Wrote:  Better solved with a calculator or pencil & paper?

http://www.telegraph.co.uk/education/edu...tears.html

If you remember how to differentiate - pencil and paper. If not, at least a scientific calculator. I solved it with num-solv on TI36X but had to put it into "classic" mode to avoid memory errors.

I think that may be the point - some calculator solvers give students errors if they rely only on the calculator.

-----------
On the other hand, it seems to me that the question wording is a bit deceiving. They state that crocodiles travel at different speeds on land versus water. Okay, Ill buy that. Then they ask the time taken for the
(ai) croc does not travel on land - I guess it swims in the water then "through the air" to the zebra? Ans: 104.4 sec
(aii) croc swims the shortest distance - Ill assume this is directly to shore then on land for the duration. Ans: 110.0 sec

So it would seem from (ai) that the swimming duration is lowest --> swimming speed is fastest. However, in (b) they ask to look for a minimum time duration between 104.4 and 110.0 sec using the given formula. Ans: x=8 at 98.0 sec. What they didn't state but may be true is that the total distance the croc travels varies based on the portion of swimming versus land crawling.

Crocodiles tend to have low physical (running) stamina and therefore would most likely only catch a zebra if it first swam right up next to it THEN came out of the river bank.

[Dieter]

(10-09-2015 03:47 PM)Gerald H Wrote:  Better solved with a calculator or pencil & paper?
http://www.telegraph.co.uk/education/edu...tears.html

So this is "...challenging, reportedly reducing some pupils to tears" for 17-18 year old students? And it is "far, far in advance of what was required"? I can't believe it.

An equation T(x) is already given, you simply have to insert two values (0 and 20) for x, and for the third question you minimize it which requires nothing but the most basic calculus skills. The solution – with paper and pencil – should not require more than a few minutes.

But maybe I do not get the point here. In this case may a native speaker please explain what's so special with this question.

BTW the answer is the number of points you can score for the correct solution. ;-)

Dieter

[Gerson W. Barbosa]

(10-09-2015 06:43 PM)Dieter Wrote:  But maybe I do not get the point here. In this case may a native speaker please explain what's so special with this question.

Dieter

Wasn't this meant for high-school students, many of those who hate math? I did it by hand in a few minutes, but I learned the technique in 1982, only when I was a Physics student.

Gerson.

[Dieter]

(10-10-2015 03:25 AM)Gerson W. Barbosa Wrote:  Wasn't this meant for high-school students, many of those who hate math? I did it by hand in a few minutes, but I learned the technique in 1982, only when I was a Physics student.

Physics? University ?-)

All it takes here is deriving a simple equation by applying the chain rule. That's something I learned back in school when I was 17. Today in many European countries students leave school at 17 or 18 (one year earlier than in the Eighties), and at this point their math skills should be sufficient for much more complex tasks than this more or less trivial crocodile problem which, as you said, can be solved in a few minutes. Even with paper and pencil. Let alone with a graphing calculator (enter equation, determine minimum) where the required time essentially depends on how fast you can type. ;-)

Dieter

[peacecalc]

Hello all,

the problem isn't to find the minimum of a given function. The problem is to find the ansatz and to derive this function.

[Paul Dale]

When my daughters needed help with their high school mathematics over the past few years, the problems were harder that this one.

Anyway, the 34S solves this easily:

Code:

01:  LBL A
02:  f'(x) B
03:  RTN
04:  LBL B
05:  x[^2]
06:  3
07:  6
08:  +
09:  [sqrt]
10:  5
11:  [times]
12:  x[<->] Y
13:  2
14:  0
15:  -
16:  4
17:  [times]
18:  -
19:  END

Then:
0 ENTER B giving 110
20 ENTER B giving 104.4
0 ENTER 20 SLV A giving 8
B giving 98.
The answer is the smallest of 110, 104.4 and 98.


- Pauli

[Thomas Klemm]

(10-10-2015 06:02 AM)Dieter Wrote:  Let alone with a graphing calculator (enter equation, determine minimum) where the required time essentially depends on how fast you can type. ;-)

Who needs a graphing calculator when you have a HP-15C?

Program for T(x)

Code:

LBL A
ENTER
ENTER
ENTER
X^2
3
6
+
SQRT
5
*
2
0
R^
-
4
*
+
RTN

Solutions to (a)

(i) crocodile does not travel on land
20
A
104.40307

(ii) crocodile swims the shortest distance possible
0
A
110.00000


Program for T'(x)

Code:

LBL B
RCL 0
I
GSB A
Re<>Im
CF 8
RCL/ 0
RTN

Initialisation
EEX
5
CHS
STO 0


Solutions to (b)

x which minimises the time taken
0 ENTER
20
SOLVE B
8.00000

minimum possible time
A
98.00000

Cheers
Thomas

[Thomas Klemm]

(10-09-2015 06:24 PM)CR Haeger Wrote:  Then they ask the time taken for the
(ai) croc does not travel on land - I guess it swims in the water then "through the air" to the zebra? Ans: 104.4 sec
(aii) croc swims the shortest distance - Ill assume this is directly to shore then on land for the duration. Ans: 110.0 sec

Quote:The time, T, measured in tenths of a second, is given by
\[
T(x)=5\sqrt{36+x^2}+4(20-x)
\]

That's maybe the tricky part?

Kind regards
Thomas

[Thomas Klemm]

(10-09-2015 04:07 PM)peacecalc Wrote:  It is the mechanical analogy of snell's law in optics, both have the same solution.

In this degenerated case for the critical angle we have:
\[
\sin \theta=\frac{v_1}{v_2}=\frac{4}{5}
\]
Assuming that the width of the river is \(d=6\) we get:
\[
x = d \tan \theta
\]

Calculation
4 ENTER
5 /
ASIN
TAN
6 *
8.0000

Cheers
Thomas

[striegel]

What kind of river has no water flow rate? I was expecting something more involved.

[Katie Wasserman]

(10-10-2015 02:56 PM)striegel Wrote:  What kind of river has no water flow rate? I was expecting something more involved.

Me too, and what kind of prey stands still? It could have been a lot more complicated and more interesting.

[Thomas Klemm]

(10-10-2015 02:56 PM)striegel Wrote:  I was expecting something more involved.

Like hunting a shark instead of a zebra?

[Thomas Klemm]

(10-10-2015 03:30 PM)Katie Wasserman Wrote:  It could have been a lot more complicated and more interesting.

Like this?

[John Colvin]

When I was in high school, the last part of the question (min. time) would have been a little more challenging considering calculus wasn't taught in high schools back then.

[brickviking]

(10-10-2015 07:14 PM)John Colvin Wrote:  When I was in high school, the last part of the question (min. time) would have been a little more challenging considering calculus wasn't taught in high schools back then.

At least in my school, I was getting taught stuff about differentiation and integration, and I'm assuming I was possibly getting taught single-variable calculus. That was in Form 6, I would have been either 16 or 17 at the time.

(Post 43)

[striegel]

(10-10-2015 03:32 PM)Thomas Klemm Wrote:  

(10-10-2015 02:56 PM)striegel Wrote:  I was expecting something more involved.

Like hunting a shark instead of a zebra?

1+
Bonus points awarded to Thomas!
(Where is that photo from, Australia?)

[Massimo Gnerucci]

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