Need help with Algebra/Geometry problem for daughter. I have solved it, but have no idea how.?

Question

OK, so here's the question: The expression 2(l+w) can be used to find the perimeter of a rectangle with a length l and width w. What are the length and width of a rectangle if the area is 13.5 square units and the length of one side is 1/5 the measure of the perimeter? Explain your reasoning.
I figured out the answer. I figured out that if two sides must be 1/5 of perimeter each and two sides must be 1.5/5 of perimeter. So I figured out that the sides must be 3,3,4.5,4.5 units each to equal a perimeter of 15. However, I just figured this out by plugging in numbers until all factors were met. I know there must be a formula to figure out this answer and I need to show my daughter that. Please help this 30+ year old who has forgotten her math knowledge! I need help with the "explain your reasoning" part. Thanks!

She is in 8th grade G/T algebra.

And she doesn't rely on her parents to do her work. It was a bonus, challenge question. I just don't know how to explain to her how to work the problem when I didn't work it the right way.

Answer 1

area of retangle = length * width
13.5= lw

Since the length is 1/5 of the perimeter,
l = 1/5{2(l+w)}
l = 2/5l + 2/5w
3/5l = 2/5w
3l = 2w
3/2l =w

Sub w=1.5l into the area equation

13.5=lw
13.5=l(1.5l)
13.5=1.5 l^2
l^2=9
l=3
since the length =3, this means the width is 1.5*3=4.5

Answer 2

Your reasoning is actually pretty good even though it may not be the way that the teacher wants her to solve the problem. If a rectangle has an area of 13.5 and the sides of the rectangle are l and w, we know that area is equal to length times width, therefore we can use the expression lw=13.5

So we know the perimeter of the rectangle in algebraic terms is 2(l+w). The problem states that the length of one side is 1/5 the measure of the perimeter. Therefore if we take 1/5 of the expression for the perimeter, we end up with (2*1/5)(l+w) or just (2/5)(l+w). We can now set one side equal to this expression we just developed; it really doesn't matter which one for the sake of our problem, but say we pick w:
w=(2/5)(l+w)

Now, solve the system of two equations in two variables:
w=(2/5)(l+w)
lw=13.5

It is best to work a bit with the top one to get it into simpler terms first:
0=2/5(l+w)-w
0=(2/5)(l)+(2/5)w-w
0=(2/5)(l)-(3/5)(w)
If we multiply everything by 5, we will have a much neater equation:
0=2l-3w
Now, solve for either variable in terms of the other.
2l=3w
l=(3/2)w

Also, it is a lot easier if you have everything in the same format (meaning you don't want decimals and fractions, you want everything in either decimal format or fraction format)
lw= 13 1/2 or lw=27/2.

If we substitute (3/2)w for l in the second equation, we can come to our solution:
(3/2)w(w)=27/2
multiply by 2 and simplify
3w^2=27, now divide both sides by 3 to isolate that w squared term:
w^2=9
Square root of 9 is 3 so if w squared is 9 then w must be 3
Going back, we know that l=(3/2)w so sub 3 back in for w
l=(3/2)(3) which equals 9/2 or 4.5

Length (l)=4.5 Width (w)=3 so the sides of the rectangle are 3, 3, 4,5, and 4.5

So, you got exactly the correct answer but just went about doing it a different way than your daughter is probably being taught at school.

Answer 3

Okay, let's start with the 2( l + w) = the perimeter of a rectangle. Make sure she understands where it comes from. Draw a picture. Show her that the base and the side parallel to it are the same so we get l + l. Same with the other two sides, so we get w +w. So the perimeter is l + l + w + w =
2 l + 2 w = 2 (l + w).
The area of every rectangle is baseXheight or lw
So lw = 13.5
The last piece of information is the hard part.
The length of one side (it doesn't make any difference so let's say w) w = one fifth the perimeter. The perimeter is 2(l + w).
So, w = 2(l + w)/5. The problem is practically solved now.
Let's simplify a little: The last eq gives us 5w = 2l + 2w or
3w = 2l or w = 2l/3. Sustitute into the area eq lw = 13.5
l (2l/3) = 13.5 or l^2 = 3*13.5/2 and l = 4.5 and from the previous equation we get w = 3.
I can't swear to the nos. but the theory is definite.
Need more help? [e-mail]

Answer 4

JM, it would help to tell us what grade your daughter is in so this won't be over her head.
Since you say algebra, I'll do the algebra version:

Start with the perimeter. If one side is 1/5 the perimeter, the opposite side is also 1/5 the perimeter. This leaves 3/5 the perimeter to be shared between the two other sides. Thus, the two other sides are 1.5 time as long as the short sides.

If X is the length of the short side, 1.5 X is the length of the longer side, and 1.5X^2= the area of 13.5 ft^2. Dividing, X^2=9 and X=3. So the length and width are 3 and 4.5.

Answer 5

>>However, I just figured this out by plugging in numbers until
>>all factors were met.

Trial and error? That's pretty bad!

>>I know there must be a formula to figure out this answer

If she comes to rely on quick formulas (or parents) to solve everything, then you won't learn how to solve things by mathematical reasoning, and she'll most likely fail her tests.

The area of a rectangle of length L and width W is just LW. We're told the area is 13.5, so LW = 13.5. We're told the length of one side (we can pick either one; swapping "length" and "width" is just turning the rectangle on its side) is 1/5 of the perimeter. So L = (1/5) * 2(L+W).

You've now got two equations with two unknowns. So solve for L and W.

Since LW = 13.5, then W = 13.5/L. Plug this into the second equation to get a single equation in terms of only L. Solve for L. Then go back and use LW = 13.5 to solve for W.

Answer 6

We have to turn the verbal information into equations.

There are two pieces of information to work with here. If the side lengths are "A" and "B", we get:

1. "the area is 13.5 square units"
means that:
AxB=13.5

2. "the length of one side is 1/5 the measure of the perimeter"
means that:
A = 2(A+B) / 5
or, more simply
5A = 2(A+B)

Here A and B are the length and width, but we don't know which is which. You need to solve the set of two equations we came up with:

AxB=13.5
and
5A = 2(A+B)

and see what you get for A and for B. The shorter one will be the width, the longer one will be the length.

Good luck.

Answer 7

ok we can do this algebraically.....
area 'a'= l * w
13.5 units² = l units * w units
perimeter 'p' = 2(l + w)

if the length of one side, no side specifically is 1/5p then we should solve for one side in terms of our perimeter equation
p = 2(1/5p + w)
p = 2/5p + 2w
p-2/5p = 2w
3/5p = 2w
w = 3/10p

now we have one side as 1/5p and the other side as 3/10p
back to our area formula
a = l * w
13.5 = 3/10p * 1/5p
13.5 = 3/50p²
p = sqrt(225) = 15

since p = 15, and we know that l = 1/5p and w = 3/10p
(in real actuality 3/10 is more than 1/5 so well reverse them here)
let: w = 1/5p and l = 3/10p

width = 1/5 * 15 = 3
length = 3/10 * 15 = 4.5

Answer 8

l * w = 13.5
We let l to be (1/5) of perimeter.
therefore l = (2/5)*(l + w)
l = (2/5)l + (2/5)w
(1-(2/5))l = (2/5)w
(3/5)l = (2/5)w
3l = 2w
w=(3/2)l
Substitute w into l * w = 13.5
l * (3/2)l = 13.5
l² = 13.5 * (2/3)
l² = 9
l = 3 (l cannot be negative)
When l = 3, w = (3/2)(3) = 4.5
Hence, the sides are 3 units and 4.5 units.

Answer 9

alright. you have to figure out what times what = 13.5 which is 3 x 4.5 (because that is how you get the area [length times width] ) after you figure that out, plug those two numbers into this equation 2(l +w) like you said. and that's how you get the answer. hope that helped. :]

Answer 10

2 ( l+ W ) = perimeter width = 1/5 ( 2L+ 2W) = 2L/5 + 2W/5

formula for area of this rectangle is

L (2L/5 + 2W/5) = 13.5

2/5 L^2 + 2/5 LW = 13.5

0.4 L^2 + 0.4 LW = 13.5

L^2 + LW = 33.75

L^2 + LW - 33.75 = 0