The area of a rectangle is
77 yd2
, and the length of the rectangle is
10 yd
less than three times the width. Find the dimensions of the rectangle.
L = 3W - 10
LW = (3W-10)W = 3W^2 - 10W = 77
3W^2 - 10W - 77 = 0
Solve quadratic equation for W, then L.
To find the dimensions of the rectangle, we'll solve the problem step by step.
Let's assume the width of the rectangle is "w" yards.
According to the problem, the length of the rectangle is 10 yards less than three times the width. So, the length would be 3w - 10 yards.
The formula to calculate the area of a rectangle is: Area = Length * Width.
Given that the area is 77 square yards, we can set up the equation as:
77 = (3w - 10) * w
Now, let's simplify the equation:
77 = 3w^2 - 10w
Rearrange the equation to bring all the terms to one side:
3w^2 - 10w - 77 = 0
Now, we have a quadratic equation. To solve it, we can factorize or use the quadratic formula. In this case, since factoring may be challenging, let's use the quadratic formula:
The quadratic formula is given by:
w = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = 3, b = -10, and c = -77.
Substituting these values into the quadratic formula:
w = (-(-10) ± √((-10)^2 - 4 * 3 * -77)) / (2 * 3)
Simplifying:
w = (10 ± √(100 + 924)) / 6
w = (10 ± √1024) / 6
w = (10 ± 32) / 6
Now we have two values for "w":
w1 = (10 + 32) / 6
w1 = 42 / 6
w1 = 7
w2 = (10 - 32) / 6
w2 = -22 / 6
w2 = -11/3
Since the width cannot be negative, we discard the negative value and accept that the width of the rectangle is 7 yards.
Now, we can find the length by substituting the width value back into the equation:
Length = 3w - 10
Length = 3 * 7 - 10
Length = 21 - 10
Length = 11
Therefore, the dimensions of the rectangle are:
Width = 7 yards
Length = 11 yards