the perimeter and area of a rectangle are 48 cm and 140cm^2 respectively. Find the length and width of the rectangle
lw = 140
2l + 2w = 48
l + w = 24
l = 24-w
in lw = 140
w(24-w) = 140
-w^2 + 24w - 140 = 0
w^2 - 24w + 140 = 0
(w-10)(w - 14) = 0
w = 10 or w = 14
if w=10, l= 140/10 = 14
if w = 14, l = 140/14 = 10
we call these symmetric roots
the rectangle is 10 cm by 14 cm
To find the length and width of a rectangle based on its perimeter and area, we can set up a system of equations.
Let's assume the length of the rectangle is L and the width is W.
Given:
Perimeter (P) = 48 cm
Area (A) = 140 cm^2
The formula for perimeter of a rectangle is:
P = 2L + 2W
The formula for the area of a rectangle is:
A = L * W
1. Using the formula for the perimeter, we have:
48 = 2L + 2W
2. Using the formula for the area, we have:
140 = L * W
We now have a system of two equations with two variables. We can solve this system of equations to find the values of L and W.
Let's solve equation 1 for L:
2L = 48 - 2W
L = (48 - 2W) / 2
L = 24 - W
Now substitute the value of L in equation 2:
140 = (24 - W) * W
Expand equation 2:
140 = 24W - W^2
Rearrange this equation to a quadratic form:
W^2 - 24W + 140 = 0
Now we can either factor this quadratic equation or use the quadratic formula to find the values of W.
Using factoring:
(W - 10)(W - 14) = 0
So, either (W - 10) = 0 or (W - 14) = 0.
If (W - 10) = 0, then W = 10.
If (W - 14) = 0, then W = 14.
Now, substitute the values of W back into equation 1 to find L:
For W = 10:
L = 24 - W = 24 - 10 = 14
For W = 14:
L = 24 - W = 24 - 14 = 10
Therefore, the possible dimensions of the rectangle are:
Length (L) = 14 cm, Width (W) = 10 cm
or
Length (L) = 10 cm, Width (W) = 14 cm.
To find the length and width of a rectangle given the perimeter and area, we can use algebraic equations.
Let's assume the length of the rectangle is represented by 'L' and the width is represented by 'W'.
We know that the perimeter of a rectangle is calculated by adding the four sides together: Perimeter = 2(L + W).
In this case, we are given that the perimeter is 48 cm, so we can write the equation as:
2(L + W) = 48.
Similarly, we know that the area of a rectangle is calculated by multiplying the length and width: Area = L * W.
In this case, we are given that the area is 140 cm^2, so we can write the equation as:
L * W = 140.
Now we have a system of two equations:
1) 2(L + W) = 48,
2) L * W = 140.
We can solve this system by substitution or elimination.
Let's use substitution:
From equation 1) we can solve for L:
L = 24 - W.
Now substitute this value of L into equation 2):
(24 - W) * W = 140.
Simplify:
24W - W^2 = 140.
Rearrange to form a quadratic equation:
W^2 - 24W + 140 = 0.
Now we can solve this quadratic equation to find the value of W. Once we find W, we can substitute it back into equation 1) to find the value of L.