Question

A homeowner wants to fence in a rectangular yard along one side of her house. The total area she can fence is 800 sq. ft
2 sides are x and one side is 60-x

What are the possible values for x that results in using the least amount of fencing around the space

And what is the least amount of fencing needed to create the 800 square foot area

Answer 1

@AZ

Answer 2

@AZ

Answer 3

@AZ

Answer 4

This is for algebra class

Answer 5

So you're not taking calculus?

Answer 6

No we’re working on quadratic equations

Answer 7

I see, okay so

this is our diagram

Answer 8

we only have three sides to fence

so the total material we need is x + x + 60-x

and we know that the total area should be 800 ft^2

Answer 9

do you know what the formula is for the area of a circle?

Answer 10

Total area to fence 800 square feet

Answer 11

we have a rectangle here with one side as 'x' and the other side as (60-x)

what would the area be?

Answer 12

x(60-x)

Answer 13

Good! And remember we said that the area is going to be 800 feet

so

x(60-x) = 800

can you multiply the x inside?

Answer 14

distribute the x into (60-x)

Answer 15

60x-x^2?

Answer 16

Good!

so we have

60x - x^2 = 800

what do you get if you subtract 800 on both sides?

Answer 17

60x-x^2-800=0

Answer 18

Good and we can rearrange that so we have

-x^2 +60x - 800 = 0

and we can solve it using the quadratic formula

\( x = \dfrac{ -b \pm \sqrt{b^2 - 4ac}}{2a} \)

when the equation is \( ax^2 + bx + c = 0\)

Answer 19

were you able to plug it into the quadratic formula? @Militza1720

Answer 20

Yes did that and I got x=20 or 40

Answer 21

Good! So now let's see which value gives us the smallest perimeter because then that means we've used the least amount of material for fencing

so remember

perimeter = x + x + x - 60
perimeter = 3x - 60

do you get a smaller perimeter when x is 20 or when x is 40?

Answer 22

20

Answer 23

what is the perimeter when x is 20?

that's the answer to your second question

Answer 24

Would it equal 0?

Answer 25

No, think about it. Would it make sense to require 0 feet of material for a fence with an area of 800?

Fencing material needed = perimeter = 3x - 60

plug in 20

3*(20) + 60 =

Answer 26

120

Answer 27

So why does -60 turn into +60

Answer 28

oh I see what you mean

Answer 29

my mistake, I accidentally wrote + instead of -

so yeah when we plug in x = 20 we see that it doesn't work out

so that means the only solution is x = 40

so plug in 40

Answer 30

Ok so I got 60

Answer 31

wait I was doing the perimeter all wrong, I'm sorry

Answer 32

perimeter = x + x + (60 - x)

perimeter = total fencing material = x + x + 60 - x

= x +60

Answer 33

I was just going off of memory so I accidentally switched the signs but do you understand now?

so x=20 would work and that would give you the smallest perimeter

Answer 34

Awesome thank you now I got it

Answer 35

No problem!

Sorry for my careless errors :)

Answer 36

No thank you for being thorough

Answer 37

Of course, it was my pleasure :)