School math question. Your input?

[MNDude]

My daughter has this problem-solving question for her homework. I'm feeling kind of dumb on this one. What do you think is the correct answer?

Mrs. Feltner wants to put a border on a baby blanket. The area of the blanket is 12 square units. Which shows how many units of materials she needs for the border?

A 12 units B 14 units C 15 units D 21 units

[FredZarguna]

14, 15, and 21 all work for the perimeter. Any number ≥ 8√3 (≈ 13.86) will work.

[TaMoDee]

See post 54. An assumption on my part that it’s 3X4.

[PAR35]

B - 14.

[pacific_waters]

You’re all wrong if it’s 3x4 then a a border of 12 units would take up the entire area so it wouldn’t be a border anymore. The correct answer isn’t there, none of the above.

[FredZarguna]

That doesn’t mean anything. You need to revisit your grade school math. Square units just means: unit x unit. That’s ALL. You can get 12 square units with a blanket 3.46 units on every side.

[TontoKowalski]

L*W = 12 and 2(L + W) = x (where x is one of the four possible values) solve first eq for W... W = 12/L substitute in second giving 2L + 24/L = x divide by 2... L + 12/L = x/2 multiply both sides by L and rearrange... L^2 - (x/2)L + 12 = 0 substituting each value (12,14,15,21) into this eq yields a quadratic that can be solved. Answer “A” has no real roots, but B,C, and D all have nondegenerate real roots, therefore B,C or D are valid answers, but not A.

I approached the problem from a "system of equations" perspective, and I agree with your analysis. My solution is that there are "infinite solutions" but that one of them is not A.

But then I realized that the problem does not specify that the blanket is rectangular, so we have to assume this in order to use our approach.

I am also frankly shocked that so many wise freepers just assume that the blanket sides must be in neat and tidy integer unit lengths, even if one concedes that the blanket must be rectangular. Nothing in the problem precludes a rectangle with a length 4.5 units, for example.

However, I believe we can safely say that the student is "supposed" to imagine a 3x4 rectangular blanket, provided the student is in a public grade school.

This thread is a candidate for the Hall of Fame. It embodies so many qualities that make Freeping a fun and unique experience.

[mmichaels1970]

If 15 and 21 work, what is L and W?

[TaMoDee]

Also see post 72.

[mmichaels1970]

Nope. 12 SQUARE units would cover the entire area. 12 units would not unless their width was the same as the quilting patches. Even so, a border extends outwards so it doesn’t matter how wide the border is.

[SpaceBar]

15 works also (L= 5.186, W=2.314)

---

You are correct.
a) Area = 12 (no solution)
b) Area = 14 (L=3, W=4)
c) Area = 15 (L=2.32, W=5.19)
d) Area = 21 (L=1.30, W=9.19)

Note roles of L and W can be reversed, and answer (b) is the only one that yields integer side lengths, but that constraint was never specified, so c and d are valid answers also. Also the above values are approximate since I just eyeballed them off a graph.

[RightOnTheBorder]

Given no shape constraints all answers except A are possible.

A right Triangle will have the smallest area to perimeter ratio and solving the quadratic for a triangle base that results in an area and perimeter of 12 yields only complex roots. Given that all other shapes will have a larger A to P ratio answer A is impossible.

Answers B,C, and D may all be achieved by assuming a rectangle and setting up two equations for each answer. 2(L)+2(W)=Answer and (L)(W)=12.

Answer B yields a 3x4 Blanket.
Answer C yields a sqrt(3/2) by 12/(sqrt(3/2)) blanket.
Answer D yields a sqrt(9/2) by 12/(sqrt(9/2) blanket.

As there is nothing that indicates the blanket dimensions must be integers they are all equally correct.

[TontoKowalski]

Baby blanket is the object ... so it’s likely to be 4x3 ... if it is 6x2, demand to see a photograph of that baby : ))

Then you are a specie-ist and should report for re-indoctrination.

I'd like to know exactly where the problem states that this blanket is for a HUMAN baby. A 6 x 2 blanket might suit nicely for a baby python.

[mmichaels1970]

Although the problem doesn’t specify that the blanket is rectangular, I think it’s safe to assume the vast majority of baby blankets are rectangular. I’m going by the fact that I had four little idiots and 100% of their baby blankets were rectangles. Until they ate the corners off.

[chalkfarmer]

Which brings us back to Mrs. Feltner and her recycling project.

[mmichaels1970]

They aren’t asking for area. They want the perimeter. So you have to have L*W=12 AND 2L+2W=one of the choices.

[mmichaels1970]

I'd like to know exactly where the problem states that this blanket is for a HUMAN baby. A 6 x 2 blanket might suit nicely for a baby python.

I'd go with 1X12 for a python.

[The_Reader_David]

Assuming the blanket is rectangular, the answer could be any of the choices except A. The perimeter of a rectangle with fixed area is minimized by a square (a simple Calculus 1 problem shows that) so for a 12 square unit rectangle, the minimum perimeter is 4 times the square root of 12, which is approximately 13.86 units. Any perimeter large than this is possible.

Even allowing a blanket of arbitrary shape, in which case a circle minimizes the perimeter (that takes calculus of variations to prove), in which case the circle has radius the square root of 12/pi and thus circumference (as the perimeter of a circle is called) 2*pi times this, which is approximately 12.28, so again A is the only answer among the choices which impossible for an arbitrarily shaped blanket.

Unless there is something in the problem you didn't tell us it is an ill-posed problem -- which is fine if it was included to make the point that not all practical problems translate into mathematical problems with unique solutions, but is horrible if some nitwit teacher is going to insist that one answer is correct because some dolt of a textbook author posed it and gave "the correct" answer in the answer key.

[Damifino]

A round blanket with a diameter of 3.908 would have an area of 12 and a perimeter of 12.2823. Just a thought..

[FredZarguna]

Solve this equations for L and W, where P is the perimeter (14, 15, 20) or any value ≥ 13.86:

L = [P ± √(P² - 2 * 4 * 24)]/4

Use the + choice in ± for L, it is usually longer. W will then be the - choice.

For example, with P = 14, L=4, W=3. With P=15, L=5.19, W=2.31. With P=20, L=8.6, W=1.39.

[Attention Surplus Disorder]

The correct answer is:

The use of the term “baby” blanket is both gender-hostile, sexist, and potentially offensive to a: infertile couples, and b: same-sex partners in states run by Republican bastards who do not believe in bestowing rights clearly granted in the Constitution to couples whose sexual and thus reproductive proclivities may extend beyond normal womb-based gestation derived from obsolete concepts left over from phallo-dominated eras of societal norms. The question, therefore, imposes a paternalistic and no doubt white-dominated orthodoxy that has an overwhelming tendency to impose mores and symbology upon those who have a proven disadvantage servicing heating bills generated by greedy and polluting utility conglomerates whose concerns are their profit levels, both their own, and their shareholders who collectively could not care less about the relative temperatures nor circulatory congruences of their customers who generally have difficulties maintaining age-appropriate body temperatures since they have been underserved and underpaid in their slave-labor jobs which invariably do not pay enough to allow them to keep up with their utility bills. The superior approach, therefore, is to allow the various stakeholders; utility company; child unit (our preferred term), parent A and parent B, garment and textile engineering contractor, to freely express their disparate motivations within the context of a regulated and orderly construct developed by enlightened, yet supportive intellects fully comprehending the full panoply of vicissitudes comprising all elements surrounding this situation both in abstract, and in concrete. Then and only then will our work be done.