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hi Wolfwood7388

Have a look at my post 12. Is this what you're asking about? In which case, are you able to say what needs more explanation.

=COUNTIF(L2:L10001,>0) this returns an error

I checked the syntax for this function and found this example:

=COUNTIF(B2:B5,">55")

Counts the number of cells with a value greater than 55 in cells B2 through B5. The result is 2.

So try adding the ""s

Bob

hi Wch12267

Welcome to the forum.

I'm going to use x rather than o here to avoid confusion with zero.

root(1-x) cannot have a real value if x is positive as the square root of a negative doesn't exist in real numbers.

So I did a bit of trial and improvement. I set up Excel with 11 lines of formulas, testing from -1 to 0 in steps -0.9, -0.8 and so on. I could see a solution lay between -0.9 and -0.8 to I repeated the search between these and gradually 'homed in' on -0.866. That looked to me like root(3)/2 so I tried that directly and the expression evaluated to 1.

Now Excel only works with 'so many' digits of accuracy so this isn't a proof of the solution, but it gave me enough to switch to an algebraic method. What I'm about to show you has to be used with caution as it can lead to values that aren't solutions as well as values that are. Let me show you why:

Suppose we have a simple equation like x = 5. That's easy to 'solve' ; the answer is 5.

But if I square it: x^2 = 25 ; this has 2 solutions, x = 5 and x = -5. The second isn't a solution of my original equation! So if you ever use this technique, always check the values you end up with, to make sure they really do solve the original problem.

I'll replace the inequality with an equals.

square both sides

Simplify and rearrange

Square again:

The 'other' value can be disregarded as it is positive and I've already discounted that.

Bob

Hi Monox D. I-Fly,

If you stand close to the tree and stand the stick so that it mimics the 'lean' of the tree (if necessary as the tree may not be vertical) and not in the tree's shadow, then the tree and its shadow and the stick and its shadow make similar triangles. So, by measuring the two shadows you can calculate the height of the tree.

Bob

hi Lauren1415

Yes, that's correct!

Bob

hi Kamov50K

Welcome. Or you could ask her to teach you

Bob

x = 6, y = 2/9 is a point of inflexion. Cannot find any others.

Bob

hi benice

Thanks beautiful! I think I may be hypnotised now.

Bob

hi phro,

Yes that first u should have been s. I refer the reader to my signature statement.

I have drawn the diagram and entered eight numbers into the regions. I still think t + w = 3. My sequence of solving is:

z, w, then t, then v, then x, then u, then y. Adding, these total to 13 as required so I'm unsure why you think there's a problem with the question.

When it says "Three of the kittens are white with spots." I interpret this as meaning the union of white and spots. I didn't assume I was being told anything about those three kittens' tails. Hence t + w.

Bob

Thanks zetafunc. It seems that the limits make all the difference.

Zeeshan 01: After a bit of research I think I can take you through this, but it'll take more than one post. If you would like that please reply. Or you could do like me and use a search engine.

Bob

hi Zeeshan 01

Wow! Where did that come from? Not out of a beginners book of integration, I hope. I couldn't see how to do it, so I used https://www.wolframalpha.com/ to compute it. It has a far from easy solution as you'll see. Did you just make this one up yourself? That's a hazard with trying that. Not all functions are easily integrable.

Bob

hi Lauren1415

Venn problems with three sets involved look like this:

So **A** might be kittens with white hair and **B** might be white spots, for example.

There are eight regions altogether : such as **s** = kittens with white hair but no spots and no long tail, and **w** = kittens with all three characteristics.

So start by drawing three overlapping circles and label on the outside "white hair" etc.

Then you have to read the information carefully so that the correct numbers go in each region. We're told that all kittens have at least one trait so **z** = zero.

And "One kitten has white hair but does not have spots or a long tail" so we can put in **s** = 1.

Some regions cannot be entered without some calculation eg. "Three of the kittens are white with spots" means they must be in **A** and also in **B**. But we don't yet know how to decide how much is **t** and how much **w**. But we do know that **t + w** = 3. Don't enter that as it's not the number for a single region.

But there's enough information to work out t and w separately, and once you have, you can enter the correct numbers. Keep doing this with all the information and you'll find that all eight numbers can be determined.

Check you've done it right by adding up numbers in regions to check it tallies with the information. All eight numbers should add up to 13 for example.

Try this and if you're still stuck post back what you think s, t, u etc are and I'll have a closer look for you.

Bob

hi Max,

Welcome to the forum.

Bob

hi Zeeshan 01

There are two stages to this problem. I think you have done the first but I'll say anyway.

This is a product of two functions so use the product rule:

You can combine the trig functions into a single trig function using the compound angle formulas. You want a cosine so I'll use

To make the derivative look like the right hand side here:

Hope that helps,

Bob

hi yorkmanz

Welcome to the forum.

Looks like you've got these sorted ; I agree with all your answers.

Bob

hi xxStellaxx

Welcome to the forum.

This question has been asked many times. Post 18 of this thread:

http://www.mathisfunforum.com/viewtopic.php?id=18391

has a diagram that may help.

Bob

hi Strangerrr

The area of the triangle will be different as it depends on both the side length and the angle at the top.

n is the number of sides. Here's an example using a regular 12 sided polygon. The centre point is O.

Let's say that A1A2 = side length = 7cm.

First I'll work out the angle A1-O-A2. Calculate 360/12 = 30.

The triangle A1-O-A2 is isosceles. The vertex angle is 30 and the base is 7.

Split it in half; see yellow shaded right angled triangle. This triangle has an angle of 15 and the 'opposite' is 3.5cm.

Use trigonometry to calculate the height.

Now calculate the area of the triangle A1-O-A2.

Now calculate the area of the whole polygon.

Hope this helps,

Bob

That's exactly what I got!

Bob

I agree with all those answers.

Provided the polygon is regular (with n sides) you can draw lines from the centre out to the vertices and make n triangles. Once you have calculated the area of one triangle you can times by n to get the area of the polygon.

These triangles will always be isosceles. The base is the side of the polygon. But you'll also need the height of the triangle.

The angle at the top of each triangle is 360/n in size. With an isosceles triangle you can always split the triangle in two down the middle, making a right angle at the bottom. The top angle will be split into two angles of 180/n each. The base is split into two parts of side/2 each. You can get the height by either

or

Then the area of the polygon is

Bob

**And a Very Happy Christmas and New Year to you!**

Bob

You're welcome!

Bob

hi Strangerrr

I'm not getting that answer. Imagine the prism made from card. If you unfold the net you'd have two triangular ends and three rectangles. The lateral area is the area of those three rectangles. Their measurements are 4 by 8, 6 by 8 and 6 by 8.

To get the volume you need the area of the triangle and then times by the height (8).

The perpendicular height of the triangle requires Pythag: sq rt (6^2 - 2^2). 2 because you make a right angle by dividing the isosceles triangle in half.

The 'slant' height isn't needed for prisms … they don't have one.

Bob

hi Spicca

Welcome to the forum.

I've been hoping that someone would post an answer to this and I could learn too. No such luck

So I'm having a go myself. Please note: I've never done this before, so what follows may be rubbish. Please comment …. ask for clarification … tell me why it's wrong etc. Maybe between us, we can arrive at the correct answer. And oh yes … your English is good.

So let's work with 3D coordinates with the x-y plane horizontal and z going straight up. Further, let's make the sphere have unit radius (cannot see any harm in that) and centred on the origin, O.

If P is one point of the tetrahedron, then we can specify its position using spherical coordinates theta and phi as shown here: https://en.wikipedia.org/wiki/Spherical … ate_system And let P' be the point in the x-y plane below P.

Now, what would be helpful is to have a formula for the volume of a tetrahedron in terms of theta and phi but I cannot find one. Plenty of internet pages giving the formula in vector terms such as https://math.stackexchange.com/question … ot-product

and as a determinant https://stackoverflow.com/questions/986 … n-4-points

I could also expand either into a large algebraic formula but it would take ages to enter all the LaTex so you'll have to ask nicely if you want this.

phi can take any random value from 0 to 2pi and theta any from 0 to pi.

So you can construct your function with 8 variables and there it is. Hhhmmmm.

Bob

hi Zeeshan 01

Sorry, but I do not understand what you are asking. Please post the whole question.

Bob

I would say this means they are the same line and intersect everywhere.

Bob

In algebra if A can be shown to be equal to B by a series of reversible algebraic steps then it follows that B is also equal to A.

Why don't you want to use the method of partial fractions ?

Bob

Why are you still posting 'help Me' questions in the 'exercises' section ?