A solid has faces that consist of 5 congruent squares and 4 congruent equilateral triangles. The solid has 9 vertices. How many edges does the solid have? (no image)
A.
16 edges
B.
18 edges
C.
20 edges
D.
22 edges

- anonymous

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- anonymous

Use\[v-e+f=2\]where v is no. of vertices, e is no. of edges, f is no of faces.

- anonymous

so 9- ? + 5 = 2 ?

- anonymous

Figure has 5 square faces PLUS 4 triangular faces, so 9 faces in total.

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- anonymous

ut oh, I'm confused.

- anonymous

What don't you understand?

- anonymous

oh wait, 9+9 is 18

- anonymous

So 18 edges, I think :o

- anonymous

Not quite. The equation is given above, and you know that f = 9 and v = 9. Are you able to solve for e?

- anonymous

I think so, because E is what I'm suppose to find out

- anonymous

That's right. Go ahead. What do you get?

- anonymous

20 edges?

- anonymous

OK. Let's take it step by step. The equation is\[v-e+f=2\]and you need to solve for e. I don't like the -e on the left hand side, so my first step would be to get of it by adding e to both sides. Can you do that?

- anonymous

so e will cancel out? v+f=2

- anonymous

But to maintain equality you have to add e to BOTH sides of the equation. So you have to fix up the right hand side.

- anonymous

I don't do too well with just variables :o

- anonymous

It's OK. Work with them just as you would work with numbers.

- anonymous

What do you get when you add e to BOTH sides of the equation?

- anonymous

v+ f(e)= 2 ?

- anonymous

No. Let's write it out. Adding e to both sides of the equation we get\[v-e+f+e=2+e\]See how we did that?

- anonymous

ooh yes okay

- anonymous

Now, we can simplify the left hand side by combining the -e and the +e. They combine to give zero, and we don't have to write +0. Some people say that they cancel each other out. Understand?

- anonymous

Yep yep, that part makes sense

- anonymous

So now we have\[v+f=2+e\]To solve for e we have to get rid of the 2 on the right hand side. How will we do that?

- anonymous

we subtract 2 from both sides, but I'm not too sure how that will look

- anonymous

That's right. Let's write it out just like before.\[v+f-2=2+e-2\]Does that make sense?

- anonymous

ohh okay, I understand

- anonymous

Almost there. Now, simplify the right hand side. What will the equation look like?

- anonymous

we subtract 2 again right?

- anonymous

Yup. There's a 2 and a -2 on the right hand side. They 'cancel each other out'.

- anonymous

so v + f - 2 = e ?

- anonymous

Perfect. Well done. Now just plug in the values for v and f and calculate the value of e. What do you get?

- anonymous

9+ 9 ? Because earlier you said 4 faces and 5 faces, which is 9 :o

- anonymous

But you're forgetting something. v=9 and f=9 so we have\[e=v+f-2\]\[e=9+9-2\]\[e=?\]

- anonymous

18-2 = 16

- anonymous

Yayyyy!

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