Question

5. Two pillars have been delivered for the support of a shade structure in the backyard. They are both ten feet tall and the cross sections of each pillar have the same area. Explain how you know these pillars have the same volume without knowing whether the pillars are the same shape.

Answer 1

To start, please state the formula for the volume of a cylinder. (which a pillar is)

Answer 2

v = (3.14)(r)^2(h)

Answer 3

Correct, so answer me this: If the pillars are of the SAME HEIGHT, and have the same cross-sectional area (meaning that they MUST have the same RADIUS), what does that tell us about the volume?

Answer 4

It means it has the same volume. To me it sounds like the shapes are congruent also

Answer 5

You're right, they are indeed congruent. So, how would you explain that they have the same volume, regardless of shape? Tell it to me like you're going to write it down please.

Answer 6

I should add that they, in a manner of speaking, must be the same shape: a cylinder.

Answer 7

The formula used to find the volume of a pillar is (v = (3.14)(r)^2(h). They are both 10 feet tall and they both have the same cross sections meaning that they have the same radius. Plugging the values into the formula gives you the same volume.

Answer 8

Nicely stated. That should satisfy your teacher.

Answer 9

thanks for the help!

Answer 10

I'd add this one thing though, change "pillar" in the first sentence to "cylinder". You're technically correct, but pillars can have odd shapes on them, making them not cylinders in some cases.

Answer 11

Ok, will do

Answer 12

Any others you need help with?

Answer 13

Um I have 3 more questions to do, this is my last assignment. I'll make a new thread so we can do 1 more?

Answer 14

Just ask it here.

Answer 15

1. A triangular section of a lawn will be converted to river rock instead of grass. Maurice insists that the only way to find a missing side length is to use the Law of Cosines. Johanna exclaims, that only the Law of Sines will be useful. Describe a scenario where Maurice is correct, a scenario where Johanna is correct, and a scenario where both laws are able to be used. Use complete sentences and example measurements when necessary.

Answer 16

Ok, so the most important thing here is the internal measurement degrees for each angle. So, to start, please state the law of Cosines and law of Sines for me.

Answer 17

Law of Sines: a / sinA = b / sinB = c / sinC
Law of Cosines: c^2 = a^2 + b^2 - 2ab cos (C)

Answer 18

Alright. Now, let's set up two sample triangles. I'll draw two possibilities right now.

Answer 19

Yea law of sines would work better

Answer 20

Now, let's consider the second triangle I drew. In that example, every interior angle could potentially be something different, and we aren't given that it's right triangle, so we don't know any of the angles to work with, unless it's given. And since we're not told if it is or not, let's assume that at least one is given. Imagining that only one interior angle & two outer sides are given, as long as they fulfill the law of cosines equation, we can use it. IN SHORT: The law of cosines equation is more useful when given two sides and an interior angle.

Answer 21

That probably sounds confusing, so please ask any questions I may have just caused you to have.

Answer 22

it was kind of confusing until i read the last sentence which makes sense. it looks like two sides and an interior angle would fit better into the cosines equation

Answer 23

so i understand

Answer 24

Ok, good. Now, the last thing you need to do is to describe a time when either could be used. This is pretty open ended, but what do you think? can you think of any time either could be used?

Answer 25

You can use sines when you have 2 side measurements and an angle measurement and you need to find the measurement opposite of the one you know.

You can use cosine when you're trying to find the 3rd side of a triangle

As for a situation for both, I have no clue

Answer 26

Alright, no problem. Just think of it this way. (Drawing in a second)

Answer 27

Yea

Answer 28

Then there ya go. You have all you need to use either sines or cosines to solve for the missing side given everything but one side.

Answer 29

Ok so I can use what I wrote and your example and that would work as my answer?

Answer 30

Ok

Answer 31

Sure, feel free. I won't tell your teacher! :)

Answer 32

Ok so would this work:

Johanna can use the Law of Sines when she has 2 side measurements and an angle measurement and she needs to find the measurement opposite of the one she already knows. Maurice can use cosine when he’s trying to find the 3rd side of a triangle. They can use both laws when they already know all of the side measurements of the triangle and they already know two angle measurements.

Answer 33

Sure, that'd work fine.

Answer 34

Well I have two more questions but theyre kind of hard to ask on here because they want you to use graphs

Answer 35

I'll post this 1 just to show

3. There are two fruit trees located at (3,0) and (–3, 0) in the backyard plan. Maurice wants to use these two fruit trees as the focal points for an elliptical flowerbed. Johanna wants to use these two fruit trees as the focal points for some hyperbolic flowerbeds. Create the location of two vertices on the y-axis. Show your work creating the equations for both the horizontal ellipse and the horizontal hyperbola. Include the graph of both equations and the focal points on the same coordinate plane.

Answer 36

I shall try my best. Let me take a look.

Answer 37

Ok, one moment. I'm going to try to start this one myself.

Answer 38

Okay

Answer 39

After some discussion, I am left with the conclusion that this is beyond my ability. What's your other problem you need to do? Perhaps I can help there.

Answer 40

If it's beyond your ability, then it's definitely beyond mine.

A pipe needs to run from a water main, tangent to a circular fish pond. On a coordinate plane, construct the circular fishpond, the point to represent the location of the water main connection, and all other pieces needed to construct the tangent pipe. Submit your graph. You may do this by hand, using a compass and straight edge, or by using GeoGebra.

Answer 41

Ok this one I don't believe is as hard, it seems like I just need to graph all of this on a coordinate plane?

Answer 42

I think you're right. It seems really open ended in that you can draw a circular drawing anywhere in a coordinate plane, with a point anywhere near it to represent the water main. After that, just draw the tangent line.

Answer 43

Ok that doesn't seem too bad but just 1 question about it. what is the tangent line?

Answer 44

A tangent line is simply one that seems to just "scrape" the edge of a circle or circle-like shape. Note that it technically never touches the circle. Here's an example. http://s3.amazonaws.com/illustrativemathematics/images/000/001/081/large/tang1_da02ab8f9928cf295c6d13e3fdc731f4.jpg?1341857748

Answer 45

Ok that makes sense.

Answer 46

I have a question as far as this last question but don't know how to phrase it haha

Answer 47

Ok so I draw the circular fish pond on a graph. Then i find a point for the water main connection. (and all other pieces needed to construct the tangent pipe) i guess that's where I'm a little lost. Is it basically doing the same thing as the picture you showed?

Answer 48

Yeah, just draw a line from the "water main" point to a point on the outer edge of the circular shape. But make sure to note that it doesn't actually touch the edge, it just gets infinitely close to it.

Answer 49

Ok so something similar to this? sorry for all these questions