Answer :
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
[tex]\frac{3l}{3}[/tex] = [tex]\frac{81}{3}[/tex]
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
[tex]\frac{24}{2}[/tex] = [tex]\frac{2w}{2}[/tex]
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:
[tex]\mathbf{2[(2w + 3) + w] = 78}[/tex]
[tex]\mathbf{length = 27 $ cm}\\[/tex]
[tex]\mathbf{width = 12 $ cm}\\[/tex]
Recall:
Perimeter of a rectangle = [tex]2(l + w)[/tex], where l is length and w is width.
Given:
length of rectangle, l, is 3 cm more than 2 times the width, w.
Thus,
- length = (2w + 3) cm
- width = w
- Perimeter = 78 cm
Equation that models this would be:
[tex]\mathbf{2[(2w + 3) + w] = 78}[/tex]
- Open bracket
[tex]2(2w + 3 + w) = 78\\\\2(3w + 3) = 78\\\\6w + 6 = 78\\\\6w = 78 - 6\\\\6w = 72\\\\\mathbf{w = 12}[/tex]
The width (w) = 12 cm
Length = (2w + 3) cm
Plug in the value of w
[tex]Length = 2(12) + 3\\\\\mathbf{Length = 27 $ cm}[/tex]
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:
[tex]\mathbf{2[(2w + 3) + w] = 78}[/tex]
[tex]\mathbf{length = 27 $ cm}\\[/tex]
[tex]\mathbf{width = 12 $ cm}\\[/tex]
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