Question

PLEASE HELP

Consider the given rectangle.

(image attached)

The perimeter is equal to y, and the area
is equal to 3y.
a) Determine equations to represent the perimeter and area.
b) Solve the system of equations algebraically.
c) Are both solutions possible? Explain.
d) State the value of x, the perimeter, and the area.

Answer 1

How do I set up the equation

Answer 2

@IMStuck

Answer 3

Sorry i was stuck

Answer 4

is it y=4x^2+10

Answer 5

For a. the perimeter is all the sides added up together. So if they are telling you that the perimeter is equal to y, then it's equal to y. Like this:
y = 2(x+5) + 2(3x)

Answer 6

If you have to expand that, it comes to y = 2x+10 + 6x...y = 8x + 10.

Answer 7

Make sense?

Answer 8

yes all add up together!

Answer 9

Ok now for the area...area is 3y. So the area of the rectangle (the formula, that is) is A = length times width. So the area is 3y...
3y = (x+5)(3x)

Answer 10

We now have two equations here, one for perimeter in terms of y and one in area in terms of y.

Answer 11

okay I'm going to write this down to make sense of things

Answer 12

\[y = 8x+10\]and \[3y=3x ^{2}+15x\]

Answer 13

Can I go on or should I wait?

Answer 14

Alrighty then...waiting

Answer 15

hold on, how did we get 3y=3x^2+15x

Answer 16

I skipped the step; should have shown you...sorry. Here it is:
3y=(x+5)(3x) expands to 3y = 3x^2 + 15x

Answer 17

Ok, so we have the perimeter equation in terms of just plain old y; so let's get the area equation in terms of just plain old y. We can do that by dividing both sides by 3. Right now we are moving on into part b.

Answer 18

This is really confusing, okay so yes i get it

Answer 19

for the i skip, let me read ur new comment

Answer 20

Do you understand how I set up the equations in the beginning?

Answer 21

I know how but I don't know why...

Answer 22

what do you mean you don't know why?

Answer 23

wait i do, its for the peremeter, and now it was for area

Answer 24

right

Answer 25

?

Answer 26

Yes. Very good. We have a perimeter equation and an area equation now. However, we want the area equation to NOT be a 3y...we want it to be a y. So we divide both sides by 3 to get the y alone.

Answer 27

So now if we devide everything by y we can have y= stuff and then we can add that into our previous equation, right

Answer 28

it becomes one equation

Answer 29

I was thinking of doing it like this...solving both for y. y = ... and y = ,,,. Then we can set
... = ,,, Like this:

Answer 30

so y=x^2+5

Answer 31

Yes, y = x^2 + 5. Now we can set that equal to 8x + 10. Like this:
x^2 + 5 = 8x + 10. Now we can solve for x.

Answer 32

uhhh

Answer 33

No you were right!!!!!

Answer 34

Ok!

Answer 35

ok now we transfer thigns

Answer 36

So let's solve it for x.

Answer 37

And don't forget you have an x^2 in there too.

Answer 38

so its x^2-8x-5

Answer 39

This will become a quadratic equation that you will have to factor to find the zeros. Yes that's right...what you wrote. Can you factor that?

Answer 40

yes i think so
_+_=-8
_x_= -5
nope i dont know

Answer 41

i should do quadratic equation?

Answer 42

most definitely!

Answer 43

quadratic formula, but yes...

Answer 44

so 8 plus minus rootsquare 44 devided by 2

Answer 45

I got something different, so let me check my answer real quick ok?

Answer 46

ok

Answer 47

I got this 3 times:

Answer 48

\[\frac{ 8\pm \sqrt{84} }{ 2 }\]which could actually reduce to\[4\pm \sqrt{84}\]S let's solve these.

Answer 49

8x8 is 64 right?

Answer 50

O Crap i think i did minus inside the square root

Answer 51

right, but you have 64 - 4 (1)(-5) and a negative times a negative is a positive, so it's 64 + 20, not 64 - 20.

Answer 52

i always do that, you have it right

Answer 53

no, you're good...you caught yourself; that's great!

Answer 54

\[4+\sqrt{84}=13.165\]

Answer 55

Can you do the other one?

Answer 56

ok i used a online quad calc to find 8.6 and -.6 rounded

Answer 57

Hmmm...I got 13.165 and -5.165

Answer 58

uhh i think its because u devided the 8 by 2, but isnt that after adding the rootsquare and 8 together

Answer 59

then deviding it by 2

Answer 60

The square root of 84 is 9.16515139. Add 4 to that and you get, rounded to the thousandths place, 13.165

Answer 61

4-9.16515139=-5.165

Answer 62

So your solutions are x = 13.165 and x = -5.165

Answer 63

Are we good with those?

Answer 64

nope, i get the same answers as the calc tho

Answer 65

i did 8+(squareroot(84)) then devide by 2 = 8.582575695

Answer 66

oh gosh, you're right...I was making an addition/subtraction error. Sorry! So sorry!

Answer 67

Okay!

Answer 68

so is it okay to round that to 8.6

Answer 69

OK, NOW we see that x = 8.583 and x = -.583, thanks to you! Phew! What an ordeal! Sorry!

Answer 70

Part C asks if both the solutions are possible. Do you know that answer?

Answer 71

and we take 8.6 because the minus makes no sense

Answer 72

That's right! Very good!

Answer 73

um its the extremenent or something rihg?

Answer 74

It's extraneous. ; )

Answer 75

The "extra" answer.

Answer 76

So now that you have that x = 8.6, you can find the perimeter and the area now, right?

Answer 77

You're pretty up on this stuff...you're really doing a great job!

Answer 78

thank you very mych, uh do i do 2(3(8.6))+ 2( 8.6+5)

Answer 79

for the perimeter, just use 8(8.6)+10

Answer 80

Remember we reduced it to y = 8x + 10?

Answer 81

fill in the x with 8.6

Answer 82

i see, but would it be okay either way?

Answer 83

yes, just a lot longer one way than the other way. But yes you would get the same answer.

Answer 84

Which is....(what's the perimeter?)

Answer 85

148.8

Answer 86

y=

Answer 87

?

Answer 88

y = 8(8.6)+10
y = 68.8+10
y = 78.8

Answer 89

oo noo

Answer 90

what?

Answer 91

ok yes makes more sense

Answer 92

got that one down now?

Answer 93

yes

Answer 94

Wow this went a long way!

Answer 95

Ok, now for the area. The formula is y = x^2 + 5. So with x being 8.6.... )I know it went a long way!) ; )

Answer 96

\[y=(8.6)^{2}+5\]What about that one?