The length of a rectangle is three centimeters more than twice the width. The perimeter is 78 centimeters. Define a variable and write and solve an equation to find out how long and wide the rectangle is?
Answer:
The rectangle is 27 centimetres long and 12 centimetres wide.
Step-by-step explanation:
To solve this problem, we need to follow the steps below;
First, we need to write down the question in a mathematical form.
Let l equal to the length of the rectangle and w equal to the width of the rectangle;
From the question; "The length of a rectangle is three centimetres more than twice the width" can be written down mathematically as
l = 3 + 2w ----------------------------------------------(1)
The question state that the perimeter is 78 centimetres.
Now the formula for calculating the perimeter of a rectangle is
perimeter = 2l + 2w
If the perimeter is 78 centimetres, this implies that;
2l + 2w = 78 ----------------------------------------------(2)
Equation (1) can be rearranged as;
l - 2w = 3 ---------------------------------------------------(3)
We can now solve equation (2) and (3) simultaneously
Add equation (2) and equation (3) together
3l = 81
Divide both-side of the equation by 3
\(\frac{3l}{3}\) = \(\frac{81}{3}\)
l = 27
substitute l=27 into equation(1)
l = 3 + 2w
27 = 3 + 2w
subtract 3 from both-side of the equation
27 - 3 = 3-3 + 2w
24 = 2w
Divide both-side of the equation by 2
\(\frac{24}{2}\) = \(\frac{2w}{2}\)
12 = w
w= 12
Therefore, the rectangle is 27 centimetres long and 12 centimetres wide.
For a rectangle whose perimeter is 78 cm, and its length is three cm more than twice the width, the equation we can derive after defining the variables is:
\(\mathbf{2[(2w + 3) + w] = 78}\)
\(\mathbf{length = 27 $ cm}\\\)
\(\mathbf{width = 12 $ cm}\\\)
Recall:
Perimeter of a rectangle = \(2(l + w)\), where l is length and w is width.
Given:
length of rectangle, l, is 3 cm more than 2 times the width, w.
Thus,
Equation that models this would be:
\(\mathbf{2[(2w + 3) + w] = 78}\)
\(2(2w + 3 + w) = 78\\\\2(3w + 3) = 78\\\\6w + 6 = 78\\\\6w = 78 - 6\\\\6w = 72\\\\\mathbf{w = 12}\)
The width (w) = 12 cm
Length = (2w + 3) cm
Plug in the value of w
\(Length = 2(12) + 3\\\\\mathbf{Length = 27 $ cm}\)
Therefore, for a rectangle whose perimeter is 78 cm, and its length is three cm more than twice the width, the equation we can derive after defining the variables is:
\(\mathbf{2[(2w + 3) + w] = 78}\)
\(\mathbf{length = 27 $ cm}\\\)
\(\mathbf{width = 12 $ cm}\\\)
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