2. A rectangle whose perimeter is 80 m has an area of 384 m^2. Find the dimensions of the rectangle. Solve it as a polynomial.
If the width is w, we have
w(40-w) = 384
w^2 - 40w + 384 = 0
since 384 = 4*96 or 8*48 or 12*32 or 16*24 - aha!
(w-16)(w-24) = 0
Now you can easily figure the dimensions. Note that width is usually considered to be less than the length.
To solve this problem using a polynomial, we need to start by identifying the dimensions of the rectangle. Let's assume the length of the rectangle is "L" and the width is "W".
We are given two pieces of information:
1. The perimeter of the rectangle is 80 m.
Perimeter = 2*(Length + Width) = 80 m
2. The area of the rectangle is 384 m^2.
Area = Length * Width = 384 m^2
Now we can set up a system of equations using these two pieces of information:
Equation 1: 2*(L + W) = 80
Equation 2: L * W = 384
To solve this system of equations, we can use the method of substitution or elimination. Here, we will use the substitution method.
First, let's solve Equation 1 for L:
2*(L + W) = 80
L + W = 40
L = 40 - W
Now, substitute the value of L in Equation 2:
(40 - W) * W = 384
Next, simplify the equation and rearrange it in the standard form of a polynomial:
40W - W^2 = 384
To solve this equation as a polynomial, we need to set it equal to zero:
W^2 - 40W + 384 = 0
This is a quadratic equation in the form of "ax^2 + bx + c = 0", where a = 1, b = -40, and c = 384. We can solve it using factoring, completing the square, or quadratic formula.
Factoring the quadratic equation, we can find:
(W - 12)(W - 32) = 0
Setting each factor equal to zero, we get:
W - 12 = 0 or W - 32 = 0
Solving for W, we find two possible width values:
W = 12 or W = 32
Now, substitute these values back into Equation 1 to find the corresponding lengths:
For W = 12:
L + 12 = 40
L = 40 - 12
L = 28
For W = 32:
L + 32 = 40
L = 40 - 32
L = 8
The two possible dimensions for the rectangle are:
Length (L) = 28 m, Width (W) = 12 m
Length (L) = 8 m, Width (W) = 32 m
So, the two possible dimensions of the rectangle are 28 m x 12 m and 8 m x 32 m.