hectictar

The diameter is 10 inches so the radius is 5 inches.

circumference of base   =   2π * (radius)   =   2π * (5 inches)   =   10π  inches

lateral area of cylinder   =   (circumference of base) * (height)   =  (10π  inches) * (21 inches)   =   210π  square inches

Assuming that the bottom of the pot is covered and the top of the pot is uncovered, we only need to add the area of one of the bases to the lateral area to find the total covered area. So let's find the area of the base and add one of those to the lateral area.

area of base   =   π * (radius)²   =   π * (5 inches)²   =   25π square inches

So...

total covered area   =   lateral area of cylinder  +  area of base

total covered area   =   210π  square inches   +   25π  square inches

total covered area   =   235π  square inches

total covered area   ≈   738.3  square inches

Here are a couple of examples:

To enter  \(2\frac34\) ,   type:     2 + 3/4

To enter  \(256\frac{15}{16}\) ,   type:     256 + 15/16

We can let the area of the entire octagon be 1. Then..

[ABCDE]   =   1/2

area of △AOB   =   1/8

area of △BOM   =   1/2 of 1/8   =   1/16

[ABMO]   =   area of △AOB + area of △BOM   =   1/8 + 1/16   =   3/16

[ABMO] / [ABCDE]   =   ( 3/16 )/( 1/2 )   =   3/8

Do you see how we know that

b  =  90° - 2a   ?

(If you're confused on that part, let us know!!)

Then once we know that is true, there are a couple ways to think of the next step...

Given that:

b  =  90° - 2a

We can take the sin of both sides of that equation to get:

sin(b)  =  sin(90° - 2a)

Also......a kind of another way to think of this....

Looking at the expression:

sin(b)

Since  b = 90° - 2a,  we can substitute  90° - 2a  in for  b  to get the following equivalent expression:

sin(90° - 2a)

And so....

sin(b)  =  sin(90° - 2a)

Let   m∠DAQ  =  a

Let   m∠PAQ  =  b

Let   m∠PAB  =  c

Notice that    △DAQ  ≅  △BAP,    and so    m∠DAQ  =  m∠PAB    which means    a  =  c

Then since the measure of an interior angle of a square is 90°...

a + b + c  =  90°

                               Since  a = c  we can substitute  a  in for  c

a + b + a  =  90°

                               Combine like terms

b + 2a  =  90°

                               Subtract  2a  from both sides to solve for  b

b  =  90° - 2a

And so....

sin(b)   =   sin(90° - 2a)

                                             Now we can use the rule that says:  sin(90° - θ)  =  cos(θ)

sin(b)    =   cos(2a)

                                             Now we can use the double-angle formula for cos

sin(b)   =   1 - 2sin²(a)

                                             And  sin(a)  =  opposite / hypotenuse  =  1 / sqrt(5)  **

sin(b)   =   1 - 2(1/5)

sin(b)   =  3/5

**If you would like more explanation on this part please feel free to ask!

(f o g)(x)   =   f( g(x) )

                                         Since  g(x) = -x + 1,  we can substitute  -x + 1  in for  g(x)

(f o g)(x)   =   f( -x + 1 )

                                         And  f(x) = x² + 2, so to find  f(-x + 1), plug  -x + 1  into

                                         the function by replacing every instance of  x  with  -x + 1

(f o g)(x)   =   (-x + 1)² + 2

                                                        Expand the squared term by multiplying it out.

(f o g)(x)   =   (-x + 1)(-x + 1) + 2

(f o g)(x)   =   x² - 2x + 1 + 2

(f o g)(x)   =   x² - 2x + 3

Can't seem to insert a newline inside the <pre> </pre>  tags.

And now I can't even edit my first response.

Anyway, here's a screenshot of my answer taken from the preview:

*edit* formatting got messed up (even though it looked fine in preview...)

IIRC, I found this forum by searching Google for "calculator". Then I noticed there was a place to ask questions, and coincidentally I also needed help on a question. I had previously tried to find help on this question through some tutoring websites, but most of those sites wanted money first, and the one site that didn't require money upfront gave me a wrong answer. So I figured I might as well try posting my question on here, and since I didn't even have to make an account or anything to post it, I figured there was no reason not to. (If I had been forced to create an account before posting, I probably wouldn't have ever posted that question, and I probably wouldn't be here now). I wasn't really expecting an answer the first time I posted a question here, since previous efforts at getting help on my question hadn't been fruitful.....so I was super surprised and happy when I received a genuine and helpful answer.

After that, I noticed a few questions that I knew how to do, and figured maybe I should try to help a few other people out, since someone helped me out first. After I answered a few questions, I thought it was kind of fun, so I stuck with it :)

Let  x  be the original price and let  y  be the price after tax.

Then...

the price after tax   =   the original price + 9% of the original price

y    =    x   +   9%  of  x

y    =    x   +   0.09  *  x

y    =    x   +   0.09x

y    =    1.09x

Then, to find the original price when the price after tax is  155.87,  plug in  155.87  for  y

155.87   =   1.09x

155.87 / 1.09   =   x

143   =   x

So the original price is  $143.00