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You are watching: How many gifts can you wrap with 25 sq ft

— Charles Hamilton Houston, NAACP legal action Director.

Introduction


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Figure 1: instances of Christmas GiftWrapping (Source).


My sister functions as an occasion planner/wedding planner. She created me an email today with the complying with gift pack question.

I have to wrap 375 boxes. 16x14x6. The wrapping file is 30 customs wide. Ns say ns need around 1100 feet. World are informing me I only need half that.

She climate referred me to a website that offers a formula for the lot of wrapping file for a given prismatic box. Her team was not acquainted with evaluating formulas, and also she request me if I can do the calculations.

This formula was supposedly created by Sara Santos, a well-known applied mathematician. However, once I looked in ~ the equation (Equation 1), i knew something was wrong. I am certain Sara derived it effectively – the difficulty is one of acquiring it ~ above the page correctly. I still mental a conversation I had actually years back with a typesetter that complained about setting mathematical type because it to be "fussy" – he charged extra for it due to the fact that it had to be done exactly right.

Here is a video that provides a good background top top the formula and how it is used.For those that don"t want the joy of computing, I include an Excel spreadsheet here to assist you. No macros, simply a simple calculator.

Background

Equation as Stated

Equation 1 is supposed to give the area that wrapping paper required for a solitary box.

Eq. 1
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The presence of the equality is what confused me. However, ns have identified what is walk on and the equality have to not be there. The first term $latex frac12cdot left( d+2cdot h+w ight)^2&s=-1$ is for rectangular boxes and also the second term $latex displaystyle 2cdot left( w+h ight)^2&s=-1$ is because that square box – the square box term is a special situation of the rectangular term with d = w.

The gift wrapping formula go not care which numbers are assigned come d, w, or h. If you have two dimensions that room equal, then make those d and also w since you have a box through a square side.

Analysis

Derivation

The derivation is straightforward given two cases: (1) square box, and (2) rectangular box. Number 3 provides my drawings of this cases. The forced wrapping file area follows directly from these drawings.

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Figure 3(a): Square-Base Box.Figure 3(b): Rectangular-Base Box.

Now I know that Equation 1 have to not contain an equality. Instead, there room two separate situations covered. Really, the square-base box case is simply a special situation of the rectangular-box case.

Wrapping record Constraint

The internet page go not point out that the derivation provides an assumption that the wrapping file is vast enough to assistance this approach to wrapping the box. To wrap a crate this means requires that the wrapping document meet the following width constraint (Equation 2).

Eq. 2$latex displaystyle w_Wrap=sqrt2cdot left( w+h ight)$square-base box
$latex displaystyle w_Wrap=fracw+d+2cdot hsqrt2$rectangular-base box

Solution to mine Sister"s Problem

My sister necessary to know how much 30-inch broad wrapping record to buy. Number 4 reflects my answer.


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Figure 4: mine Answer to my Sister"s Problem.


1100 feet of wrapping paper is a the majority of paper.

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Conclusion

I love to job-related geometric problems, and also my sister offered me a helpful one – everyone has to wrap packages. This problem also fits in nicely with my understanding in origami and file engineering.