The length of a rectangle is
8 ft
less than three times the width, and the area of the rectangle is
35 ft2
. Find the dimensions of the rectangle.
Let's assume the width of the rectangle is x ft.
According to the given information, the length of the rectangle is 8 ft less than three times the width. Therefore, the length can be expressed as (3x - 8) ft.
The area of a rectangle is given by the formula: Area = length x width.
Substituting the given values into the formula, we can write the equation:
35 = (3x - 8) * x
Expanding and rearranging the equation, we get:
35 = 3x^2 - 8x
Now, let's solve this quadratic equation for x using factoring or quadratic formula.
3x^2 - 8x - 35 = 0
Factoring the quadratic equation, we can rewrite it as:
(3x + 7) * (x - 5) = 0
Setting each factor equal to zero, we get two possible values for x:
3x + 7 = 0 or x - 5 = 0
Solving these equations, we find:
3x = -7 => x = -7/3
x = 5
Since the width cannot be negative, we discard the negative value of x.
Therefore, the width of the rectangle is 5 ft.
Now, we can substitute this value of x into the equation for the length:
Length = 3x - 8
= 3 * 5 - 8
= 15 - 8
= 7 ft
Thus, the dimensions of the rectangle are: width = 5 ft and length = 7 ft.
To find the dimensions of the rectangle, we can set up two equations based on the given information.
Let's assume:
Width of the rectangle = W ft
Length of the rectangle = L ft
1. The first equation is based on the given information that the length of the rectangle is 8 ft less than three times the width:
L = 3W - 8
2. The second equation is based on the given information that the area of the rectangle is 35 ft²:
Area of rectangle = Length × Width
35 = L × W
Now we have a system of two equations:
L = 3W - 8
35 = L × W
To solve this system of equations, we can substitute the value of L from the first equation into the second equation:
35 = (3W - 8) × W
Expanding the equation:
35 = 3W² - 8W
Rearranging the equation to standard quadratic form:
3W² - 8W - 35 = 0
Now we have a quadratic equation. We can solve it by factoring or using the quadratic formula. Let's use factoring:
(3W + 7)(W - 5) = 0
Setting each factor equal to zero and solving for W:
3W + 7 = 0 or W - 5 = 0
Solving these equations:
3W = -7 W = 5
W = -7/3
Since width cannot be negative, we disregard W = -7/3.
Therefore, the width of the rectangle is W = 5 ft.
Substituting this value back into the first equation:
L = 3(5) - 8
L = 15 - 8
L = 7
Therefore, the length of the rectangle is L = 7 ft.
So, the dimensions of the rectangle are:
Width = 5 ft
Length = 7 ft
if the width is w, then
w(3w-8) = 35
Hint: 35=5*7