This has been bugging me for a while. Say you are doing some home improvements. You want to expand your 12 x 12 bedroom a bit so you bump it out a bit and add three feet along one wall. Your significant other then says they can't live without a new closet, so you build a three foot deep closet along the newly shorter of the walls. Basically your room has gone from 12 x 12 or 144 sq ft to 9 x 15 or 145 sq ft.

My question is, if the linear distance is the same, why isn't the area?

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Originally posted by Mike ZeidlerThis has been bugging me for a while. Say you are doing some home improvements. You want to expand your 12 x 12 bedroom a bit so you bump it out a bit and add three feet along one wall. Your significant other then says they can't live without a new closet, so you build a three foot deep closet along the newly shorter of the walls. Basically your room has gone from 12 x 12 or 144 sq ft to 9 x 15 or 145 sq ft.

My question is, if the linear distance is the same, why isn't the area?

Well, you're going to hate the obvious answer, but: Linear distance and area are not related.

Taken to an extreme, a 1 x 23 hallway would ALSO have the same linear distance, but only 23 ft².

You can probably do some fancy stuff with hypotenuses in there too and maybe even get a better answer, but I'm not bored enough.

EDIT: Or "perimeter/area ratio"

(This isn't "One Question..." material - moved to Random)

If you want to maximize area, you need to make a circular room. A circular room with a circumference of 48 feet (which is the same distance around as your 12 x 12 room) would have about 183 sq ft.

Back when I was a Calc 1 tutor, this type of question was referred to as 'the hump." Once you could get through these types of relationships and work them forwards and backwards, you had very little trouble with anything else in the course. It's one of the most basic application of calculus, but requires a fair bit of work to intuitively understand.

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The initial problem is stated incorrectly. Where you have "along the newly shorter of the walls", you meant to say "along the now *longer* of the walls.

You started with 12 x 12 = 144

you added 3 feet, along one 12 foot wall, that's 36 additional sqft; hence 144 + 36 = 180 = 12 x 15

then you deducted 3 feet along the now longer wall; that's 3x15 for a 45 sqft deduction; meaning 180 - 45 = 135 = 9 x 15

You added 36 sqft, but then took away 45 sqft.

----------- If you had built the closet "along the newly shorter of the walls", i.e. the 12 foot wall, then you'd be taking away 36, & would be back to 144 sqft. Or, if you'd built a 3x12 closet along part of the 15 foot wall, you'd be back to 144 sqft.

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