A rectangle has a width that is 3 less than twice the length. If the rectangle has an area of 170 square inches, what is the length of the rectangle?
PLEASE HELP
check your post of 1:05 pm before reposting, eh?
Length = L.
Width = 2L-3.
A = L * W = 170.
L * (2L-3) = 170.
2L^2 - 3L - 170 = 0,
L = (-B +- Sqrt(B^2-4AC))/2A.
L = (3 +- Sqrt(9 + 1360))/4,
L = (3 +- 37)/4 = 0.75 +- 9.25 = 10, and -8.5. In.
Use positive value: L = 10 In.
To find the length of the rectangle, let's assign a variable to represent it.
Let's say the length of the rectangle is "L".
According to the problem, the width is 3 less than twice the length.
So, the width would be 2L - 3.
The area of a rectangle is equal to its length multiplied by its width.
Area = Length × Width
In this case, the area is given as 170 square inches.
So we can write the equation:
170 = L × (2L - 3)
Now, let's solve this equation to find the value of L.
Expanding the equation:
170 = 2L^2 - 3L
Rearranging the equation by moving all terms to one side:
2L^2 - 3L - 170 = 0
We have a quadratic equation. To solve it, we can use factoring, completing the square, or the quadratic formula.
In this case, factoring may not yield nice whole number solutions. So, let's use the quadratic formula:
L = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 2, b = -3, and c = -170.
Substituting these values into the quadratic formula:
L = (-(-3) ± √((-3)^2 - 4(2)(-170))) / (2(2))
L = (3 ± √(9 + 1360)) / 4
L = (3 ± √(1369)) / 4
Now, let's simplify this further:
L = (3 ± 37) / 4
This gives us two potential solutions for L:
L1 = (3 + 37) / 4 = 40 / 4 = 10
L2 = (3 - 37) / 4 = -34 / 4 = -8.5
Since the length cannot be negative in this context, the length of the rectangle is 10 inches.
To solve this problem, we need to set up an equation relating the width and length of the rectangle using the given information.
Let's assume the length of the rectangle is represented by the variable "l". According to the problem, the width is 3 less than twice the length. We can express this as:
Width = 2l - 3
The area of a rectangle is given by the formula:
Area = Length x Width
We are given that the area is 170 square inches, so we can write:
170 = l * (2l - 3)
Now, we can solve this equation to find the length of the rectangle.
First, distribute the "l" to the terms inside the parentheses:
170 = 2l^2 - 3l
Rearrange the equation to set it equal to zero:
2l^2 - 3l - 170 = 0
To solve quadratic equations like this, we can either factor it or use the quadratic formula. In this case, the equation cannot be easily factored, so we will use the quadratic formula:
l = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 2, b = -3, and c = -170. Plugging these values into the formula, we get:
l = (-(-3) ± √((-3)^2 - 4(2)(-170))) / (2*2)
= (3 ± √(9 + 1360)) / 4
= (3 ± √1369) / 4
Now, let's simplify further:
l = (3 ± √1369) / 4
We have two possible solutions: one with the positive square root and the other with the negative square root. However, we are dealing with the length of a rectangle, so negative values are not meaningful in this context. Therefore, we can ignore the negative square root and focus on the positive square root:
l = (3 + √1369) / 4
Finally, we evaluate the square root:
l = (3 + 37) / 4
= 40 / 4
= 10
Therefore, the length of the rectangle is 10 inches.