THE LENGTH OF A RECTANGLE IS 3M MORE THAN 2 TIMES ITS WIDTH. IF THE AREA IF THE RECTANGLE IS 99CM^2, FIND THE DIMENSIONS OF THE RECTANGLE TO THE NEAREST THOUSANDTH.
If L = length and W = width,
L = 2W + 3 and LW = 99
Substitute and solve for either L or W.
We can substitute the value of L from the first equation into the second equation:
(2W + 3)W = 99
Distribute and simplify:
2W^2 + 3W = 99
Rearrange the equation to make it a quadratic equation:
2W^2 + 3W - 99 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:
W = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 2, b = 3, and c = -99. Substituting these values into the quadratic formula:
W = (-3 ± √(3^2 - 4(2)(-99))) / (2(2))
Simplifying further:
W = (-3 ± √(9 + 792)) / 4
W = (-3 ± √801) / 4
To find the approximate values of W, we need to evaluate the square root:
W ≈ (-3 + √801) / 4 ≈ 5.39
W ≈ (-3 - √801) / 4 ≈ -11.89
Since the width cannot be negative, we can discard the negative value of W. Therefore, the width is approximately 5.39.
To find the length (L), we can substitute this value back into the first equation:
L = 2W + 3 ≈ 2(5.39) + 3 ≈ 13.77
Therefore, the dimensions of the rectangle to the nearest thousandth are width = 5.39 m and length = 13.77 m.
To find the dimensions of the rectangle, we can set up two equations based on the given information:
1. The length of the rectangle is 3m more than 2 times its width:
L = 2W + 3
2. The area of the rectangle is 99cm^2:
LW = 99
We can substitute the value of L from the first equation into the second equation to get:
(2W + 3)W = 99
Now we can solve this equation to find the value of W, which represents the width of the rectangle.
Expand the equation:
2W^2 + 3W = 99
Rearrange the equation into a quadratic form:
2W^2 + 3W - 99 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring is not straightforward, so let's use the quadratic formula:
The quadratic formula is:
W = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 2, b = 3, and c = -99. Substituting these values into the quadratic formula, we have:
W = (-3 ± √(3^2 - 4 * 2 * -99)) / (2 * 2)
Simplifying the expression under the square root:
W = (-3 ± √(9 + 792)) / 4
W = (-3 ± √801) / 4
Approximating the square root of 801:
W ≈ (-3 ± 28.301) / 4
Using both the positive and negative square root:
W ≈ (25.301 / 4) or (-31.301 / 4)
Simplifying further:
W ≈ 6.325 or -7.825
Since a negative value for the width doesn't make sense in this context, we discard W = -7.825.
So, the width of the rectangle is approximately 6.325 m.
Now, substitute this value of W back into the first equation to find the length L:
L = 2W + 3
L = 2(6.325) + 3
L = 12.65 + 3
L ≈ 15.65
Therefore, the dimensions of the rectangle are approximately 15.65 m (length) and 6.325 m (width) to the nearest thousandth.