Question

flvs help not understanding 6.09
Option 2 - Rectangular Box

Answer 1

You will need the following materials to find the volume of a rectangular box:

A rectangular box such as a cereal or shoe box
Ruler or tape measure
Graphing technology (e.g., graphing calculator or GeoGebra)
Procedure:

Measure and record the length, width and height of the rectangular box you have chosen in inches. Round to the nearest whole number.

Apply the formula of a rectangular box (V = lwh) to find the volume of the object.

Now suppose you knew the volume of this object and the relation of the length to the width and height, but did not know the length. Rewriting the equation with one variable would result in a polynomial equation that you could solve to find the length.

Rewrite the formula using the variable x for the length. Substitute the value of the volume found in step 2 for V and express the width and height of the object in terms of x plus or minus a constant. For example, if the height measurement is 4 inches longer than the length, then the expression for the height will be (x + 4).

Simplify the equation and write it in standard form.
Find the solutions to this equation algebraically using the Fundamental Theorem of Algebra, the Rational Root Theorem, Descartes' Rule of Signs, and the Factor Theorem.

(Hint: If the numbers are large, graph the function first using GeoGebra to help you find one of the zeros. Use that zero to find the depressed equation which can be solved by factoring or the quadratic formula.)

Substitute 0 for the function notation and, using graphing technology, graph the function.

Answer the following questions
What does the Fundamental Theorem of Algebra indicate with respect to this equation?
What are the possible rational solutions of your equation?
How many possible positive, negative and complex solutions are there in your equation?
Graph the function. What type of function has been graphed (linear, quadratic, cubic, or quartic)? Provide your reasoning and describe the end behavior of the graph.
How do the solutions of the equation compare to the length of the rectangular object, and the x-intercept of the graph? Provide both the solutions and measurement.

Send to your instructor a lab report with the following information:
Title(1 point)
Materials Used (1 point)
Procedure(1 point)
Data (20 points)
Include:
A description of the rectangular box chosen (1 point)
The length, width, height and volume of the object (3 points)
The equation for the volume of the object written in terms of x (2 point)
The graph of the function (2 point)
The solutions of the equation including the algebraic work used to find the solutions (6 points)
Answers to the five questions (6 points)
Conclusion(2 points)
What did you think of the project?
What did you learn?
Do you have any questions or concerns?

Answer 2

What part are you not understanding? The lab project as described is basically:

Find a box.
Measure the box.
Do some math on the measurements.
Record everything.

Which part are you stuck at?

Answer 3

i got my measurements l=8 w=5 h=3 i just need help getting it started

Answer 4

"Apply the formula of a rectangular box (V = lwh) to find the volume of the object. "

V=lwh means V = l times w times h, so that is what you do next.

Answer 5

i got 120

Answer 6

OK. So write about it. Answer all the stuff at the bottom with that.

Answer 7

Well, that is part of what is at the very bottom.

Answer 8

but first i need this to do that well some of them
Rewrite the formula using the variable x for the length. Substitute the value of the volume found in step 2 for V and express the width and height of the object in terms of x plus or minus a constant. For example, if the height measurement is 4 inches longer than the length, then the expression for the height will be (x + 4).

Answer 9

Yep.... All sorts of fun. so your l was l=8, which is longer than w=5 and h=3. So you need to change it so x\(\pm\)?=8 to find those things.

Answer 10

Theirs example is something 4 inches longer than the length would be (x+4). In your case, the length is the longer one... that means the others would both be shorter.

Answer 11

so like 120=x(x+5)(x+3)

Answer 12

Well, let me work with what they did to find those.

Lets say in the example, h = 7 and it is 4 longer than l so l = 3, but how would I use those to find (x+4)?

Well, take x + h - l
x + 7 - 3
x + 4

In your case, l=8 w=5 h=3

So what are
x + h - l
and
x + w - l

Answer 13

so x+3-8

x+5-8

Answer 14

And simplify.

Answer 15

x-5
x-3

Answer 16

So those are the two terms you nees to use in your equation.

Answer 17

so 120= x(x-5)(x-3) right???

Answer 18

Yes.

Answer 19

Now, do you know how to get that into standard form?

Answer 20

kind of

Answer 21

Multiply it all out.

Answer 22

x^3-8x^2+15x ?

Answer 23

Yep, \(x^3-8x^2+15x\)

Now you need to look at a whole bunch of questions...

What does the Fundamental Theorem of Algebra indicate with respect to this equation?

What are the possible rational solutions of your equation?

How many possible positive, negative and complex solutions are there in your equation?

Graph the function. What type of function has been graphed (linear, quadratic, cubic, or quartic)?

Provide your reasoning and describe the end behavior of the graph.

How do the solutions of the equation compare to the length of the rectangular object, and the x-intercept of the graph?

Answer 24

for the first one is because it has 3 roots 0,3,5

Answer 25

o and for 2

Answer 26

Well, you multiplied it out... because it had 3 roots, it ended up as a 3dr degree equation. So it works both ways. If it is 3rd degree it has 3 roots. because you multiplied 3 roots you got 3rd degree.

Answer 27

and for the second one it would be the roots i would get 0,
3,5

Answer 28

Yah, since you multiplied it, kind of hard not to get the same result.

Answer 29

Now for the all possible question, you need to use the Rational Root Theorem

Answer 30

1 positive and 2 negative idk

Answer 31

i dont know how to do this one well

Answer 32

Hmmm... to be honest, this is one place where the separation fromt he directions is not great.... #2 could be answere done of two ways, with the answers or with the Rational Root Theorem list of possibilities by factoring...

For #3, I mean tot say Descartes' Rule of Signs... that is where you look for changes in the sign. You use that method to answer #3?

Answer 33

Two sign changes.
http://hotmath.com/hotmath_help/topics/descartes-rule-of-signs.html

Answer 34

o ok

Answer 35

Because of the + out front, which is normally not written, I got 2. And since it is 2 or less by 2, I get 2 or 0. See how I did that?

Answer 36

yes

Answer 37

On the lower part of the page it talks about how to find the possible number of negatives. Do you understand how they are changing signs with (-x) ?

Answer 38

u make all the x negative

Answer 39

-x^3+8x^2-15x ?

Answer 40

It has to do with putting (-x) into the function and seeing what happens because of the powers.

For odd powers, like \(x^5\), \(x^3\), \(x\) aka: \(x^1\) the sign changes happen. \((-2)^3\) means (-2)(-2)(-2) or -8. \((-x)^3\) means (-x)(-x)(-x) or -x^3.

For even powers, the power erases the - part. \((-2)^2=(-2)(-2)=4\) and \((-x)^2=(-x)(-x)=x^2\)

So odd powers flip sign, but even powers do not.

And I say flip sign because a - become + and a + becomes -.

Answer 41

so was i right
all the signs changed

Answer 42

The even power does not change sign.

Answer 43

so the 8 would stay negative

Answer 44

Yes.

\(x^3-8x^2+15x\implies \)

\((-x)^3-8(-x)^2+15(-x)\implies \)

\(-x^3-8x^2-15x\)

Answer 45

so it would be 2 or 0 again because we just changed two

Answer 46

No. You don't count how many you changed. You evaluate the new equation dor how many changes it has in it.

There are three things and it goes from negative to negative to negative. How many times does it change in there?

Answer 47

3?

Answer 48

Nope. If I say negative positive negative, my word changed twice. Once from negative to positive and once from positive to negative.

If I say positive positive positive, they are all positive and there are no changes.

So, if I say negative negative negative, it is how many changes?

Answer 49

no changes there all the same

Answer 50

Exactly. =) so 0. Or because we are evaluating negative roots, 0 possible negative roots.

Answer 51

how would i do Graph the function. What type of function has been graphed (linear, quadratic, cubic, or quartic)?

Answer 52

Well, first, all the linear, quadratic, cubic, or quartic things depend on the highest power. So what is the highest power in yours and what word does it match up with?

Answer 53

like the highest number?

Answer 54

The highest power. \(\Large x^n\) whichever n is biggest.

Answer 55

so x^3

Answer 56

Yes, is that linear, quadratic, cubic, or quartic?

Answer 57

cubic

Answer 58

Yep. So there is the answer to that part.

As for graphing it, that depends on how you have been taught to graph things.

Answer 59

not really taught how to just i do it idk

Answer 60

Well, there are several possible ways to graph. You can find lots of points. You can do the zeros and then see what is to each side, high or low, and later you learn to use calculus to find the high and low points. But for most algebra classes, it is usually the zeros and test points between to see if things go high or low.

Answer 61

So your zeros were 0, 3, and 5. Start by putting points on the x axis for those. Then take one thing lower than 0 and see if it goes higher or lower. -1 is fine. Then something between 0 and 3. Then between 3 and 5. Then above 5.

Answer 62

sorry im a visual learners im kind of getting it