How is this incorrect? I found the pentagon area and then multiplied by the height? | Mathematics | Page 1

nova temple Mar 11, 2025, 9:14 PM

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I got 0.22579 as the apothem and 17.5 as the perimeter and once you multiply those together and divide it by 2, you get the area of the pentagon which is 1.975686 and then you multiply that by the height, 9, and that's how I rounded to the nearest tenth and got 17.8

stone tendonBOT Mar 11, 2025, 9:14 PM

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nova temple I got 0.22579 as the apothem and 17.5 as the perimeter and once you multiply tho...

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limpid narwhal Mar 11, 2025, 9:16 PM

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nova temple I got 0.22579 as the apothem and 17.5 as the perimeter and once you multiply tho...

Don't do any intermediate calculations.
Take the side of the base as a and the height as h, then express the volume V in terms of a and h, and only then substitute the values.

nova temple Mar 11, 2025, 9:17 PM

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Huh?

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The area of a pentagon is 1/2 ( apothem x perimeter )

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Not whatever that is

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Area of a prism = Bh ( B = area of base )

limpid narwhal Mar 11, 2025, 9:17 PM

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nova temple The area of a pentagon is 1/2 ( apothem x perimeter )

If you want to triangulate it, don't forget to multiply the result by 5.

nova temple Mar 11, 2025, 9:17 PM

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What does it mean to triangulate

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I'm trying to get the figure on screen

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Not a triangle

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Do you want me to go step by step and you can see where I messed up?

limpid narwhal Mar 11, 2025, 9:19 PM

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nova temple What does it mean to triangulate

Split into triangles. So, like this.
Based on your approach, I assumed that's what you did.

limpid narwhal Mar 11, 2025, 9:19 PM

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nova temple Do you want me to go step by step and you can see where I messed up?

Sure.

nova temple Mar 11, 2025, 9:19 PM

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Yeah that's what I did

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Okay so

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We know the Volume of a prism is Bh

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Pentagon Area = 1/2(apothem x perimeter )

limpid narwhal Mar 11, 2025, 9:19 PM

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If by B you mean the area of the base, then yes.

nova temple Mar 11, 2025, 9:19 PM

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Yeah

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So I split the pentagon into 5 triangles

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Then split one of the triangles to get this figure

limpid narwhal Mar 11, 2025, 9:19 PM

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Right.

nova temple Mar 11, 2025, 9:20 PM

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Each inscribed angle is 72 but since we cut it to find the apothem, it's half

limpid narwhal Mar 11, 2025, 9:20 PM

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Yes.

nova temple Mar 11, 2025, 9:20 PM

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The perimeter of each side is 3.5/2 is 1.75

limpid narwhal Mar 11, 2025, 9:20 PM

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Well, length, not perimeter.

nova temple Mar 11, 2025, 9:20 PM

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Then I did the tan(36)=1.75/x

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Yeah

limpid narwhal Mar 11, 2025, 9:20 PM

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Right.

nova temple Mar 11, 2025, 9:20 PM

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and then I multipled both sides by x

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to get x(tan(36))=1.75

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then divded to get x=1.75/tan(36)

limpid narwhal Mar 11, 2025, 9:21 PM

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Fine so far, though you missed the units and the degree sign.

nova temple Mar 11, 2025, 9:21 PM

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Those aren't needed rn right?

limpid narwhal Mar 11, 2025, 9:21 PM

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If you want to use values, which you generally shouldn't do, they are always needed.

nova temple Mar 11, 2025, 9:22 PM

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Oh okay

limpid narwhal Mar 11, 2025, 9:22 PM

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Better to solve this as usual - generally.

nova temple Mar 11, 2025, 9:22 PM

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I plugged this into my calc

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and got this

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limpid narwhal Mar 11, 2025, 9:22 PM

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No, wait.

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That's not a good idea.

nova temple Mar 11, 2025, 9:22 PM

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Wdym

limpid narwhal Mar 11, 2025, 9:22 PM

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Don't do any intermediate calculations.

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That introduces error.

nova temple Mar 11, 2025, 9:22 PM

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So I should keep it as is?

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1.75/tan(36)

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For the apothem

limpid narwhal Mar 11, 2025, 9:22 PM

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Yes.

nova temple Mar 11, 2025, 9:22 PM

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Okay

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So

limpid narwhal Mar 11, 2025, 9:22 PM

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Oh, hold up.

limpid narwhal Mar 11, 2025, 9:23 PM

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nova temple

That isn't correct. You forgot the degree sign, and that affected the answer.

nova temple Mar 11, 2025, 9:23 PM

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Oh I don't have to do that with my hand held calculator

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I used a random one online

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Let me get my hand held and see if I get the same answer

limpid narwhal Mar 11, 2025, 9:23 PM

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Then use radians.

nova temple Mar 11, 2025, 9:24 PM

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Oh

limpid narwhal Mar 11, 2025, 9:24 PM

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Anyway, that's for later.

nova temple Mar 11, 2025, 9:24 PM

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I failed the radians unit

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I wasn't here the entire unit

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No idea how to do it

limpid narwhal Mar 11, 2025, 9:25 PM

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The apothem has length x = a/(2tan(π/5)). What next?

nova temple Mar 11, 2025, 9:25 PM

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I thought it was 1.75/tan(36)

limpid narwhal Mar 11, 2025, 9:25 PM

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Yes. Which is a/(2tan(π/5)) for a = 3.5 in.

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And with the correct angular units.

nova temple Mar 11, 2025, 9:25 PM

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Oh shoot on my Ti84 I got 2.40866

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Way different than what was on the website

limpid narwhal Mar 11, 2025, 9:26 PM

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Yes, that will be right. Though, we don't need that value right now.

nova temple Mar 11, 2025, 9:26 PM

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ok

limpid narwhal Mar 11, 2025, 9:26 PM

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nova temple Pentagon Area = 1/2(apothem x perimeter )

We found the apothem, so let's continue with this.

nova temple Mar 11, 2025, 9:26 PM

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So 1.75/tan(36) x 17.5

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Is the area of the pentagon

limpid narwhal Mar 11, 2025, 9:26 PM

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Yes, 5a^2/(4tan(π/5)). That's the area of the base.

nova temple Mar 11, 2025, 9:27 PM

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Then multiply that by 9

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For the prism

limpid narwhal Mar 11, 2025, 9:27 PM

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Right. So, V = 5a^2 h/(4tan(π/5)).

nova temple Mar 11, 2025, 9:27 PM

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Then divide by 2

limpid narwhal Mar 11, 2025, 9:27 PM

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No, why?

nova temple Mar 11, 2025, 9:27 PM

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I forgot to divide by 2 earlier

limpid narwhal Mar 11, 2025, 9:28 PM

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Ah. Yeah.

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See, that's why it's better to solve generally.

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Just numbers won't show where you made a mistake.

nova temple Mar 11, 2025, 9:28 PM

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What step did I forget to divide by 2

limpid narwhal Mar 11, 2025, 9:28 PM

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That's what I'm talking about.

limpid narwhal Mar 11, 2025, 9:28 PM

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nova temple So 1.75/tan(36) x 17.5

Here.

nova temple Mar 11, 2025, 9:28 PM

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The area of a pentagon is 1/2(perimeter)(apothem)

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ah

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so that divided by 2 gives us 189.7 rounded to the tenths place

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Okay so in general I just made a calculator mistake.

limpid narwhal Mar 11, 2025, 9:29 PM

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Well, yeah, the degrees problem was the most important one.

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And, again, you should solve problems generally, if possible.

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Do you want me to show how I'd write it?

nova temple Mar 11, 2025, 9:29 PM

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I know how to write them

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It's just akward

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To write such big equations that I can js solve

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But I get the reason why

limpid narwhal Mar 11, 2025, 9:30 PM

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I mean how to write it properly.

nova temple Mar 11, 2025, 9:30 PM

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Oh, sure! I'd like a little lesson.

limpid narwhal Mar 11, 2025, 9:32 PM

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Anyway, like this.
V(prism) = S(base)h
S(base) = (1/2)P(base)x = (5/2)ax = (5/2)a*(a cot(π/5)/2) = (5/4)a^2 cot(π/5)
V(prism) = (5/4)a^2 h cot(π/5) = (5/4)(3.5 in)^2*(9 in)*cot(π/5) = 189.7 in^2

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So, the approach for solving most math, physics and chemistry problems is the following:

Introduce variables for the given quantities and for what you need to find.
Derive the formula for what you need to find in terms of what you're given.
Subsitute the values of given quantities.

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Note that we don't do any intermediate calculations. At least, when possible.

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That way we don't introduce rounding errors.

nova temple Mar 11, 2025, 10:07 PM

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ooh okay

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Sorry I was doing something but I had to read it for a sec

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I'll try to do this so I don't have to round my problerms

limpid narwhal Mar 11, 2025, 10:10 PM

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Alright, nice!

whole anchorBOT Mar 12, 2025, 11:00 PM

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@nova temple

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#How is this incorrect? I found the pentagon area and then multiplied by the height?