Question 1171116: Hi
Andrew was given a rectangular cardboard 28cm by 16cm. How many right angle triangles 4cm high base 3cm could cut out.
Thanks
Found 3 solutions by MathLover1, greenestamps, ikleyn:
Answer by MathLover1(20849)   (Show Source):
Answer by greenestamps(13198)  (Show Source):
You can put this solution on YOUR website!
The response from the other tutor is certainly not correct. You can't determine the number of triangles simply by dividing the area of the whole sheet of cardboard by the area of one of the triangles. When you actually cut out the triangles, there will (with the given dimensions of the piece of cardboard) be some cardboard that can't be cut into triangles.
Two of the right triangles together make up a 3x4 rectangle. If one dimension of the original cardboard were a multiple of 4 and the other a multiple of 3, then all the cardboard could be cut into the triangles.
But neither of the given dimensions is a multiple of 3; so there will be some cardboard that can't be used to make the triangles.
Both dimensions are multiples of 4.
Consider the number of 3x4 rectangles we can make from a piece of cardboard that is 28x16.
(1) If the long dimension of each small rectangle is in the same direction as the LONGER dimension of the piece of cardboard, then we can get 28/4 = 7 rectangles in that direction. Then we could get 16/3 = 5 rectangles (remember it has to be a whole number) in the other direction. That would give us 7*5=35 rectangles, each consisting of 2 of the triangles, for a total of 70 triangles.
(2) If the long dimension of each small rectangle is in the same direction as the SHORTER dimension of the piece of cardboard, then we can get 16/4 = 4 rectangles in that direction. Then we could get 28/3 = 9 rectangles (remember it has to be a whole number) in the other direction. That would give us 4*9=36 rectangles, each consisting of 2 of the triangles, for a total of 72 triangles.
So the latter method of cutting the triangles from the piece of cardboard gives us more triangles.
It is remotely possible, however, that some unusual arrangement of the triangles on the piece of cardboard would allow us to squeeze out one or even two more triangles; so we can't be completely sure of our answer.
ANSWER: A maximum of 72 right triangles with height 4cm and base 3cm can be cut from a piece of cardboard that is 28cm by 16cm.
Answer by ikleyn(52775)  (Show Source):
You can put this solution on YOUR website! .
Andrew was given a rectangular cardboard 28cm by 16cm. How many right angle triangles 4cm high base 3cm could cut out.
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Two right angled triangles with the legs of 3 cm and 4 cm, placed hypotenuse to hypotenuse, form a 3x4-rectangle.
Let's calculate how many such rectangles can be placed onto the 28x16 cm cardboard.
We have two basic placements.
One placement is to direct the 4-cm side of the rectangle along its 28 cm dimension.
By doing this way, we have 28/4 = 7 rectanles in this direction and, OBVIOSLY, 5 rectangles in the perpendicular direction,
(because 16/3 = 5.33 = 5 when rounded to the closest smaller integer).
In all, we have 7*5 = 35 rectangles and, hence, 35*2 = 70 right angled triangles at such placement.
Next, consider another placement, directing the 4-cm side side along the 16-cm side of the cardboard.
We have then the 3-cm side along the 28-cm side of the cardboard.
By doing this way, we have 4 rectangles along the 16-cm side of the cardboard and, OBVIOUSLY, 9 rectangles along its 28 cm side
(because 28/3 = 9.33 = 9 rounded to the closest smaller integer).
In all, we have 4*9 = 36 rectangles and, hence, 36*2 = 72 right angled triangles at such placement.
Of these two opportunities, we chose the placement, which gives maximum numer of rectangles (36) and maximum number of triangles (72).
ANSWER. Maximum number of triangles is 72.
Solved.
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Such problems teachers give to young (advanced ?) students of 5th - 6th grades to check if they are able
to think on self-standing basis and to teach them to solve such problems accurately (and to think accurately)
on the self-standing manner.
The solution by @MathLover1, based on consideration the areas ONLY, gives an ESTIMATION ONLY
for the maximum number of triangles from the top, but does not give the exact number of triangles.
The solution by @MathLover1 IS NOT what is expected.
Learn on how to solve such problems (and how to teach your students, if you are a teacher) - from my post.
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