FIND THE LENGTH AND WIDTH?
A GARDEN IS SHAPED LIKE A RECTANGLE WHOSE PERIMTER IS 84 FT. LENGTH IS 2 FT. MORE THAN WIDTH.
↓ l ↓
↓ w w ↓
↓ l ↓
P = l + w + l + w = 2 · l + 2w if l = 2 + w , then
P = 2 · (2 + w) + 2w = 4w + 4 = 4(w + 1) = 84, then
w + 1 = 84/4 --------> w +1 = 21 -----> w = 20 ft. and l = 22 ft.
To answer this question we need to know a few key concepts.
- A rectangle has 4 sides.
- The perimeter of a rectangle is equal to : 2L (the length of 2 sides) + 2W (the width of 2 sides)
- Therefore the formula for the perimeter of a rectange is : P = 2L + 2W
We will use these concepts to from an equation based on what we know.
- We know that the perimeter is 84 ft.
- We know that the length is 2 ft more than the width.
We will let length = 2L
We will let width = 2W
Therefore our equation will be: 2L + 2W = 84
Since we know that the length is 2 ft more than the width , then we know that L= W +2
We can substitute W+2 for L in our equation.
Therefore our equation becomes: 2(W+2) + 2W = 84
We can simplify our equation using the distributive property.
Our equation becomes: 2W +2W +4 -84 or 4W + 4 = 84
We want to isolate our unknown variable (4L) on one side of the equation.
To isolate 4L we will subtract 4 from both sides of the equation.
4W + 4 -4 = 84-4 (Remember whatever opearation we perform on one side of the equation, we must perform the same identical operation on the other side fo the equation).
Our eqation becomes 4W =80
To solve for W(width) we divide both sides of the equation by 4.
4W/4 = 80/4
W = 20
Therefore L = W +2 = 20 +2 or 22..
The width is 20 ft.
The length is 22 ft.
The final step in solving any equation is to substitute the values into our equation to ensure they satisfy it.
Our equation P(perimeter) = 2L (length) + 2W (width)
2(22) + 2 (20) = 84
44 + 40 = 84
Our solution satisfies our equation, (formula).
The garden is 22 ft. long and 20 ft. wide.
We know there are 4 sides, totaling an 84 foot perimeter.
Carry the 4 across the 0, and flip the sign
therefore L=20 and W=20+2=22.
It's like this: You know that the length of this rectangle and the width of this rectangle put together will add up to 42. Remember, the perimeter of the whole thing is going to be four different sides because it's a rectangle. The whole perimeter is 2l + 2w. Or, say, length plus width plus length plus width. Since length equals length and width equals width, you can just cut it all in half: 2l + 2w = 84 is the same as saying l + w = 42.
That makes it easier, huh? Now let's look at this equation: l + w = 42.
I don't know what either l or w is. However, I do know that l is 2 GREATER THAN w. Now is where I use a very valuable tool you have surely heard of: algebra. Let's say that w is a variable I'll call X. Now, if w is X, I can come up with an expression for l, because I know l is 2 more than w. Therefore, l is (x + 2).
Therefore, the equation l + w = 42 can be re-written as: (x + 2) + x = 42.
This simplifies to 2x + 2 = 42
This simplifies to 2x = 40 because we subtract 2 from each side.
Then, dividing by 2 we get that x = 20.
But wait, hold your horses. Don't forget what X is. You could still miss this problem if you, say, thought X was length. It is not. X is width. Length is greater than width by 2, remember?
Therefore, the width is 20 and the length is 22. This is a 20 by 22 garden.
I hope you like that. :-)
First, we need to name our variables:
w = Width
l = Length
Next, we are going to write equations from the information given.
The formula for the perimeter of a rectangle is 2 times the length plus 2 times the width, so the formula for the perimeter of the garden is:
2l + 2w = 84
The problem also tells us that the length is 2 more than the width, which produces the equation:
l = w + 2
Since we have a system of two equation in two variables, we solve them by substitution to find their values.
2l + 2w = 84 Formula for the are of the garden
2(w + 2) + 2w = 84 Substitution
2w + 4 + 2w = 84 Distribute the 4
4w + 4 = 84 Simplify
4w + 4 - 4 = 84 - 4 Subtract 4 from each side
4w = 80 Simplify
4w/4 = 80/4 Divide both sides by 4
w = 20 Simplify
l = w + 2 Second equation from the problem
l = 20 + 2 Substitution
l = 22 Simplify
The garden is 22 ft long and 20 ft wide.