the length of a rectangle is 1 cm less than twice its width. the area is 28 cm squared. find the dimensions of the rectangle.
Just translate the English into "math"
"the length of a rectangle is 1 cm less than twice its width"
l=2w-1
"the area is 28"
w(2w-1) = 28
solve as a quadratic.
Hint: it factors very nicely
width = 4
length + 7
To solve this problem, we'll start by setting up the equation based on the given information.
We know that the length of the rectangle is 1 cm less than twice its width, so we can write the equation as:
l = 2w - 1
The area of a rectangle is given by the formula:
Area = length * width
We're given that the area is 28 square cm, so we can write the equation as:
w * (2w - 1) = 28
Now, let's solve this equation by factoring.
To factor the quadratic equation w * (2w - 1) = 28, we need to express it in the form:
(ax + b)(cx + d)
To find a, b, c, and d, we can expand the expression on the right side:
w * (2w - 1) = 28
2w^2 - w = 28
2w^2 - w - 28 = 0
Now, we need to find two numbers whose product is -56 (the product of a and d) and whose sum is -1 (the coefficient of the linear term, -w). After some trial and error, we find that the numbers are -8 and 7:
2w^2 - 8w + 7w - 28 = 0
(2w^2 - 8w) + (7w - 28) = 0
2w(w - 4) + 7(w - 4) = 0
(w - 4)(2w + 7) = 0
Setting each factor to zero gives us two possible values for w:
w - 4 = 0 (w = 4)
2w + 7 = 0 (w = -7/2) - Discard this solution since the width cannot be negative.
So, the width of the rectangle is 4 cm.
We can substitute this value back into the equation for the length to find its value:
l = 2w - 1
l = 2 * 4 - 1
l = 8 - 1
l = 7 cm
Therefore, the dimensions of the rectangle are a width of 4 cm and a length of 7 cm.
To solve this problem, we need to translate the information given into mathematical equations.
Let's say the width of the rectangle is represented by the variable "w". According to the given information, the length of the rectangle is 1 centimeter less than twice its width. So, we can express the length in terms of "w" as (2w - 1).
The area of a rectangle is given by multiplying its length and width. In this case, we are told that the area is 28 square centimeters. So, we can set up the equation:
w(2w - 1) = 28
Now, we can solve this equation to find the value of "w". This equation is quadratic, but it factors very nicely.
By factoring, we can write the equation as (2w - 4)(w + 7) = 0
Setting each factor equal to zero, we get two possible solutions:
2w - 4 = 0 or w + 7 = 0
Solving the first equation, we find:
2w = 4
w = 2
Solving the second equation, we find:
w = -7
Since we're dealing with a rectangle, the width cannot be negative, so we discard the solution w = -7.
Therefore, the width of the rectangle is 2 cm.
Now, we can substitute this value back into the length equation to find the length:
l = 2w - 1
l = 2(2) - 1
l = 4 - 1
l = 3
Therefore, the dimensions of the rectangle are:
Width = 2 cm
Length = 3 cm