Question 1202194: The area of a rectangle is 45m^2, and the length of the rectangle is 1m more than twice the width. Find the dimensions of the rectangle.
Found 4 solutions by mananth, Theo, ikleyn, josgarithmetic:
Answer by mananth(16946)   (Show Source):
You can put this solution on YOUR website! The area of a rectangle is 45m^2, and the length of the rectangle is 1m more than twice the width. Find the dimensions of the rectangle.
Let width be x m
length will be as per the info given 2x+1
Area of rectangle = L * width
x(2x+1) =45
2x^2 +x =45
2x^2+x -45=0
2x^2+10x-9x -45=0
2x(x+5)-9(x+5)=0
(x+5)(2x-9)=0
x=-5 or x= 4.5
x cannot be negative
So x=4.5
Width = 4.5 m
Length = 2x+1 = 2(4.5)+1= 10
10*4.5 =45
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! area of rectangle = 45 square meters.
the length of the rectangle is 1 meter more than twice the width.
L = length
W = width
L = 2W + 1
A = area = 45
A = L * W
replace L with 2W + 1 to get:
A = (2W + 1) * W
simplify to get:
A = 2W^2 + W
since A = 45, equation becomes:
45 = 2W^2 + W
subtract 45 from both sides of the equation to get:
2W^2 + W - 45 = 0
factor to get:
(2W + 10) * (W - 4.5) = 0
solve for W to get:
W = -10 or W = 4.5
W can't be negative, so W = 4.5
since A = 45 and W = 4.5, then L has to be equal to 10.
L = 2W + 1 becomes 10 = 2*4.5 + 1 which becomes 10 = 10, confirming the relationship between L and W is correct.
solution is that the dimensions of the rectangle are 10 by 4.5
10 is the length.
4.5 is the width.
Answer by ikleyn(52775)  (Show Source):
You can put this solution on YOUR website! .
The area of a rectangle is 45m^2, and the length of the rectangle is 1m more
than twice the width. Find the dimensions of the rectangle.
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I will show you a simple way to solve the problem MENTALLY.
Let W be the width, in meters; then the length is (2W+1) meters.
The area equation is
W*(2W+1) = 45.
Multiply both sides by 2. You will get
2W*(2W+1) = 90. (1)
So, the product of two numbers, (2W) and (2W+1), is 90, and the numbers differ by 1.
At this point, you can easily guess the solution: it is 2W = 9 and W = 9/2 = 4.5.
Next, left side of equation (1) is monotonically increasing function for positive W.
It means that equation (1) has a UNIQUE solution in positive numbers, which we just found.
ANSWER. The problem has a unique solution: the width is 4.5 m; the length is 2*4.5+1 = 10 m.
Solved.
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My reasoning in this solution is mathematically and logically absolutely strict,
although it includes an element of guessing; but this guessing is supported
by another reasoning, which shows that the guessed value is unique.
It is much easier than to factor a quadratic equation on the way
(which you practically will not be able to factor mentally without an error, in any case).
Such approach works for many other similar problems.
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About factoring, I always repeat my standard joke:
it works especially good, when you know an answer in advance.
Answer by josgarithmetic(39616)  (Show Source):
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