I can't seem to figure this out. It is for my study guide. Please help.
The length of a rectangle is 5 feet less than twice the width. The area is 25 sq feet. Using w as the variable, write an equation that can be used to calculate the width. Then find both the width and length of the rectangle.
Is the equation. 2w-5^2? How would I find both the width and length?
let the width be w
length = 2w-5
w(2w-5) = 25
2w^2 - 5w - 25 = 0
(2w+5)(x-5) = 0
w = -5/2 or w = 5, but the width could not be negative, so
the width is 5 and the length is 2(5)-5 or 5
your rectangle is a square.
To solve this problem, we can start by setting up the equation based on the given information.
Let's use w as the width of the rectangle. According to the problem, the length is 5 feet less than twice the width. So, the length can be expressed as (2w - 5).
The area of a rectangle is calculated by multiplying the length by the width, which in this case is given as 25 sq feet. So we have the equation:
(w)(2w - 5) = 25
Now, let's solve this equation to find the value of w, which represents the width.
Distribute the w into the parentheses:
2w^2 - 5w = 25
Move all the terms to one side to form a quadratic equation:
2w^2 - 5w - 25 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring is not straightforward, so we'll use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / (2a)
For our quadratic equation, a = 2, b = -5, and c = -25.
Substituting these values into the quadratic formula:
w = (-(-5) ± √((-5)^2 - 4(2)(-25))) / (2(2))
Simplifying:
w = (5 ± √(25 + 200)) / 4
w = (5 ± √225) / 4
w = (5 ± 15) / 4
This gives us two possible values for w:
w1 = (5 + 15) / 4 = 20 / 4 = 5
w2 = (5 - 15) / 4 = -10 / 4 = -2.5
Since the width cannot be negative (as it represents a physical measurement), we disregard w2.
Therefore, the width of the rectangle is 5 feet (w = 5).
To find the length, we substitute the value of w = 5 into the expression we derived earlier for the length:
Length = 2w - 5 = 2(5) - 5 = 10 - 5 = 5
So, the length of the rectangle is also 5 feet.
To summarize, the width and length of the rectangle are both 5 feet.