The length of a rectangle is 3 centimeters less than four times it’s width. It’s area is 10 square centimeters. Find the dimensions of the rectangle.
L = 4 w - 3
L w = 10 so L= 10/w
10 /w = 4 w -3
10 = 4 w^2 - 3 w
4 w^2 - 3 w - 10 = 0
w = 2 works, other root is negative
then L = 10/2 = 5
Let's assume the width of the rectangle is w centimeters.
According to the given information, the length is 3 centimeters less than four times the width, which can be expressed as 4w - 3.
The formula for the area of a rectangle is length × width. So, we can set up the equation:
(w)(4w - 3) = 10
Simplifying this equation will help us solve for the dimensions.
4w^2 - 3w = 10
Rearranging the equation to bring everything to one side:
4w^2 - 3w - 10 = 0
Now, we can solve this quadratic equation to find the values of w.
Using factoring, the equation can be factored as:
(4w + 5)(w - 2) = 0
Setting each factor equal to zero:
4w + 5 = 0 or w - 2 = 0
Solving for w:
For 4w + 5 = 0:
4w = -5
w = -5/4
For w - 2 = 0:
w = 2
Since the width cannot be negative in this context, we can disregard the negative solution.
Therefore, the width of the rectangle is 2 centimeters.
To find the length, we can substitute this width value into the equation for the length:
Length = 4w - 3
Length = 4(2) - 3
Length = 8 - 3
Length = 5
Thus, the dimensions of the rectangle are 2 centimeters for the width and 5 centimeters for the length.
To find the dimensions of the rectangle, we need to solve the given problem step by step. Let's break it down:
Let's assume the width of the rectangle as 'w' centimeters.
According to the problem, the length of the rectangle is 3 centimeters less than four times its width.
So, the length = 4w - 3 centimeters.
The area of a rectangle is equal to its length multiplied by its width.
Given that the area is 10 square centimeters, we can set up the equation:
Area = Length * Width
10 = (4w - 3) * w
Now, we can solve this quadratic equation:
10 = 4w² - 3w
Rearranging the equation:
4w² - 3w - 10 = 0
We can either factorize this equation or solve it using the quadratic formula. Let's use the quadratic formula:
w = (-b ± √(b² - 4ac)) / (2a)
For our equation, a = 4, b = -3, and c = -10.
Substituting these values into the formula:
w = (-(-3) ± √((-3)² - 4 * 4 * (-10))) / (2 * 4)
w = (3 ± √(9 + 160)) / 8
w = (3 ± √169) / 8
Simplifying further:
w = (3 ± 13) / 8
So we have two possibilities for the value of w:
w₁ = (3 + 13) / 8 = 16 / 8 = 2
w₂ = (3 - 13) / 8 = -10 / 8 = -5/4 (reject since width cannot be negative)
Therefore, the width of the rectangle is 2 centimeters.
Now, we can find the length using the length = 4w - 3:
Length = 4 * 2 - 3 = 8 - 3 = 5 centimeters.
Hence, the dimensions of the rectangle are width = 2 centimeters and length = 5 centimeters.
L = 4W - 3
L * W = (4W-3) * W = 10
Solve for W, then L.