请教數學題

jilin1960

Thirty-two feet of fencing materials is used to enclose a garden that is a rectangluar region .if the numble of square feet in the region is f ,what is the largest possible value of f?

jasonsuen

32/4 = 8
f = 8*8 = 64
square is a kind of rectangle.

jilin1960

Thx.孩子以為rectangle 不可以是square.

kenwong888

jilin1960 發表於 15-2-14 21:07
Thirty-two feet of fencing materials is used to enclose a garden that is a rectangluar region .if th ...

It is nothing dealing with rectangle or square?
What is the reason square is max area?
Formally we use differentiation to solve the problem:-
f= (16-x)x
df/dx = 16-2x

For local max, df/dx=0
so x=8

right???

Unclejt

Completing the square:

(16-x)x = 64 - (64 - 16x + x^2) = 64 - (x - 8)^2 >= 64,

and equality holds when x = 8.

kenwong888

Unclejt 發表於 15-2-15 16:24
Completing the square:

(16-x)x = 64 - (64 - 16x + x^2) = 64 - (x - 8)^2 >= 64,

Without the loss of generality, I don't think this method is good enough!

Let say if the total length is 31 right???

fungsir

if it is a rectangle, the largest possible value of f should be 63.

kenwong888

fungsir 發表於 15-2-15 22:48
if it is a rectangle, the largest possible value of f should be 63.

Interesting?
f needs to be integer right?
Anyway...just kidding...

Unclejt

回覆 kenwong888 的帖子

If the total length is A, then
(A/2 - x)x = A^2/16 - (A/4 - x)^2.
Maximum value = A^2/16, attended at x = A/4.

dlee_ca

原帖由 jasonsuen 於 15-02-14 發表
32/4 = 8
f = 8*8 = 64
square is a kind of rectangle.

Square is kind Rectangle.  Interesting.

kenwong888

Unclejt 發表於 15-2-15 22:58
回覆 kenwong888 的帖子

If the total length is A, then

The completing square method really depends on how the student can confirm the problem is quadratic at first glance right???
But I do think the student can easily classify the problem as finding local max or min right???

1234ats

Whether a square is a rectangle or not depends on the definition of these shapes.
My son’s primary school and textbook teach students that squares are not rectangles.
But if one defines a rectangle as a quadrilateral with equal opposite sides and all the four angles equal to 90 degrees, then squares should be considered as rectangles.
I would say both are correct but students should realize their differences.
In this maximum area problem, apparently square is rectangle. Otherwise the maximum area will be infinitesimally  close to 64 but not equal to 64, which is an alien concept for those who have not learnt calculus.

1234ats

Kenwong888

I would expect anyone who can master quadratic equations should be able to instantly identify that X(16-X) is a quadratic expression. It is equal to -(X-0)(X-16).The minus sign shows this parabolic curve has a maxima. And this maxima vertex is located at X = (0+16)/2 = 8 [i.e. mid way between 0 & 16] because of the symmetrical nature of parabola. Thus one should be able to get maximum area 8x8=64 by heart. It is a few-second-problem.

But anyway, I think this problem is a bit too hard for primary students. It would be more suitable to post it in secondary school section.

jasonsuen

講真，這是一道小學的數學題，但屬思考性題目，請用一個小學生能明白的方法吧，否則就是不切實際。

kenwong888

1234ats 發表於 15-2-16 10:04
Kenwong888

I would expect anyone who can master quadratic equations should be able to instantly ide ...

Exactly true!

laorenjia

妻子話我頭腦簡單，channel 同細路仔差唔多，若我講數細路仔聽唔明，就冇人會講得個細路明。（其實係見氹我教阿女數，等佢唔駛出馬）

若果呢題係小學數，就咪鬼講乜嘢completing the square, 細路仔點會明？你同個細路講:正方形係矩形一種，若周界一樣，正方形就面積最大。唔駛嚴格證明，呢條題目32/4=8。8 x 8 就最大面積，叫細路試吓7 x 9，6 x 10...就知面積越嚟越細。反之9 x 7, 10 x 6... 亦然，搞掂。

JustAParent

原帖由 laorenjia 於 15-02-17 發表
本帖最後由 laorenjia 於 15-2-17 16:10 編輯

妻子話我頭腦簡單，channel 同細路仔差唔多，若我講數細路 ...

根本就是一條簡單得很的題目。有時老師只是叫學生隨便唱兩句歌，結果家長請個樂團伴奏。

kenwong888

laorenjia 發表於 15-2-17 15:59
妻子話我頭腦簡單，channel 同細路仔差唔多，若我講數細路仔聽唔明，就冇人會講得個細路明。（其實係見氹我 ...

I really do not understand your logic!
Maybe modern people need more training!

laorenjia

kenwong888 發表於 15-2-17 23:56
I really do not understand your logic!
Maybe modern people need more training!

都話妻子氹我啦。唔用代數或微積分講得你明，唯有另請高明。

王家爸爸

1234ats 發表於 15-2-16 10:03
Whether a square is a rectangle or not depends on the definition of these shapes.
My son’s primary  ...

What is a rectangle?  A rectangle is a parallelogram with four right angles.
What is a square? A square is also a parallelogram with four right angles.
Therefore, a square is a special case of rectangle which has all sides of the same length.