Math question...?

Question

Ok, I've tried this but I'm really not confident that I did it right. The question is..

A rectangle has a perimeter 20 m. Express the area of the rectangle as a function of the length of one of its sides.

I'm guessing there is more than one answer to this question.

Can someone please help me out?

Thanks so much!

Answer 1

Define our symbols:
P := perimeter
l := side length
w := other side length
A := area

Formulas that we know:

P = 2*(l + w)
A = l*w

Rearranging the first equation:
l = P/2 - w

Substituting into the second equation:

A = (P/2 - w)*w

We are given that P = 20, substituting:

A = (20/2 - w)*w
A = (10 - w)*w

Answer 2

The area of any rectangle is given by the two variable function

A(l, w) = l * w

and the perimeter of any rectangle is given by the two variable function

P(l, w) = 2 l + 2 w,

where l is the length of the triangle, and w the width of the triangle.

Now we know the length of one of it's sides, meaning we either know l or w. and we also know that

P(l, w) = 40.

Assume, without loss of generality, that we know the length, and not the width. We use the variable s to represent the legnth of the side we know. Why can we assume we just know the length? It's because the formulas for area and perimeter are *symmetric* in their two variables meaning A(l, w) = A(w, l) and P(l, w) = P(w, l) , which holds due to the commutative properties of multiplication and addition respectively.

So now we want to find A(s):

P(s, w) = 40 = 2 s + 2 w
<=>
2 w = 20 - 2 s
<=>
w = 10 - s
<=>
A(s, w) = s * w = s * (10 - s) = 10 s - s^2

Now notice that A(s, w) is independent of w, so we now ahve a function only of the length of one of its sides, hence:

A(s) = 10 s - s^2

If you want to check to see if this works if we assume the known side was w, and we called the known width by the variable s, instead here goes:

P(l, s) = 40 = 2 l + 2 s
<=>
2 l = 20 - 2 s
<=>
l = 10 - s
<=>
A(l, s) = l * s = (10 - s) * s = 10 s - s^2.

Once again, we have A(s) = 10 s - s^2, as A(l, s) is independent of s. So we were right in asserting that it doesn't matter whether we consider the length or the width to be known. Another reason as to why it doesn't matter which you choose is because the distinction between the two is largly arbitrary, switching the labels between the two should never really change anything.

Answer 3

Not perimeter is the distance around the rectangle so
a+b+a+b=20
2a+2b=20
So not that a*b=ab which is the area
To express area to one side i.e side a
Then is
ab/a

Answer 4

let L = length , W = width
rectangle of perimeter is P = 2L+2W = 2(L+W) = 20
divide both sides by 2
L+W= 20/2 = 10
you can solve for L in term of W or solve for W in term of L
because L+W =10
if you want to solve for L so you need to cancel W, the only ways to cancel W is subtract W both sides so L+W-W = 10-w
L= 10-W cuz W- W is 0, that will be the same with solving W in term of L subtract both sides by L
L= 10 -W, or W =10-L
area of rectangle is A= LW
replacing W= 10-L above to A= LW
A=LW = L(10-L) if they given length
same here replacing L =10- W
A= LW= (10-W)W if they given width.
2 answer for thatquestion A= (10-W)W or A = L(10-L)

Answer 5

the perimeter of a rectangle is 2*(length+width).
So in your case, 2 * ( length + width ) = 20,
length + width = 10
length = x
width = 10-x

10-x
________
|________| x

Area of a rectangle is length TIMES width. So take your length (x) and your width (10-x) and multiple them.
A=x(x-10)

Answer 6

The perimeter is 20 m therefore 2(l+b)=20
l+b = 10
l= 20 - b
Therefore area is l*b
A = l*(10-l)
(Here l is length and b is width)

Answer 7

yah of coarse

The perimeter is 20 m so 2(a+b)=20
a+b = 10
a= 20 - b
so area is a*b
A = a*(10-l)
(Here a is length and b is width)

Answer 8

let the sides be a and b
so, 2(a+b)=20
or a+b=10

or a=10-b

so area =ab=b(10-b)=10b-b^2

Answer 9

hang on

call the "one side" .......... "x"

since perimeter = 20, and two of the sides have value "x",
then the other two sides add up to (20 - 2x)

one of the "other sides then is (10 -x)

AREA = x * (10 -x)
Area = 10x - x^2

max area .. = when dA/dx = 0
10 - 2x = 0
10 = 2x
x = 5

MAX area when sides are 5,5,5 & 5... a square .... who'd have thought ?\

lolol

Answer 10

x= length of one side

2x= length of one side and its parallel side.

20-2x= combined length of other side and its paralel.

20 = 2x+(20-2x)