Ellipse

**rtblue** · November 25th 2010, 08:04 AM

Horace is an avid fisherman whose favorite fishing spot is a perfectly elliptical pond with a maximum width of 144 meters and a maximum length of 240 meters. Horace has found the best spots for fishing in this pond are at the foci of the pond. So one day he sets off from the bank of the pond and rows directly to the closest focus. Finding no fish at this spot, he rows to the second focus, only to discover he has left the rest of his bait back on the bank at his original starting point on the bank. He rows back to the original starting point to pick up his bait. How many meters has Horace rowed?

That is my problem. I drew the ellipse as they said i should. The major axis as 240 and the minor axis as 144 meters long. That means that a=120, and b=72. I solved for c and got c=96. When this is multiplied by two, this should give the distance between the foci (96*2=192). Where should i go now? Any help is appreciated.

**skeeter** · November 25th 2010, 09:52 AM

Originally Posted by rtblue

Horace is an avid fisherman whose favorite fishing spot is a perfectly elliptical pond with a maximum width of 144 meters and a maximum length of 240 meters. Horace has found the best spots for fishing in this pond are at the foci of the pond. So one day he sets off from the bank of the pond and rows directly to the closest focus. Finding no fish at this spot, he rows to the second focus, only to discover he has left the rest of his bait back on the bank at his original starting point on the bank. He rows back to the original starting point to pick up his bait. How many meters has Horace rowed?

That is my problem. I drew the ellipse as they said i should. The major axis as 240 and the minor axis as 144 meters long. That means that a=120, and b=72. I solved for c and got c=96. When this is multiplied by two, this should give the distance between the foci (96*2=192). Where should i go now? Any help is appreciated.

check out this cool online construction of an ellipse ... it should give you an idea towards a solution.

How to construct (draw) an ellipse - Math Open Reference

**Soroban** · November 25th 2010, 02:20 PM

Hello, rtblue!

Horace is an avid fisherman whose favorite fishing spot is a perfectly elliptical pond
with a maximum width of 144 meters and a maximum length of 240 meters.
Horace has found the best spots for fishing in this pond are at the foci of the pond.
So one day he sets off from the bank of the pond and rows directly to the closest focus.
Finding no fish at this spot, he rows to the second focus, only to discover he has left
the rest of his bait back on the bank at his original starting point on the bank.

He rows back to the original starting point to pick up his bait.
How many meters has Horace rowed?

We are expected to know the geometric characteristics of an ellipse.

Center this ellipse at the origin.

We have: . $a = 120,\:b = 72$
The foci are: . $c^2 \:=\:a^2-b^2 \:=\:120^2-72^2 \:=\:9216 \quad\Rightarrow\quad c \:=\:\pm 96$

Code:

                       72♥
                         |        P
                         |        o(x,y)
                         |    *    *
                         |*         *
                      *  |           *
                  *      |            * 
          F2  *          |             *F1
  - ♥ - - o - - - - - - -+- - - - - - - o - - ♥ - -
  -120   -96             |             96    120
                         |
                         |
                         |
                         |
                      -72♥
                         |

Horace rowed from $\,P$ to $F_1$ , then $F_1$ to $F_2$ , then $F_2$ back to $\,P.$

We know that $\overline{F_1F_2} = 192.$

What about $\overline{PF_1} + \overline{PF_2}$ ?

Given any point $\,P$ on an ellipse, the sum of the distances from $\,P$
. . to the two foci is a constant, namely, $2a.$

. . That is: . $\overline{PF_1} + \overline{PF_2} \:=\:2(120) \:=\:240$

Therefore, he rowed: . $192 + 240 \:=\:432\text{ meters.}$

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