The 3 by 3 grid below shows nine 1 cm × 1 cm squares and uses 24 cm of wire. What length of wire is required for a similar 20 by
20 grid?
1 answer:
Answer:
840
Step-by-step explanation:
Hi there!
Usually, this kind of problem asks us the perimeter rather than the area. But when talking on this grid on how many cms are necessary to construct one made up of 9 squares 1cm x1 cm. We need to check one thing: Some line segments will be used twice. So this is not a classical perimeter question.
1) So we either manually and carefully count it avoiding not doing it twice the same common line segments.
Or.
2) By just applying an algorithm, Where n is the <em>nxn</em> grid
So, for the 20x20 grid
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Answer:
The integral for the arc of length is:
By using Simpon’s rule we get: 1.5355453
And using technology we get: 2.3020
The approximation is about 33% smaller than the exact result.
Explanation:
The formula for the length of arc of the function f(x) in the interval [a,b] is:
We need the derivative of the function:
And we need it squared:
Then the integral is:
Now, the Simposn’s rule with n=4 is:
In this problem:
So, the Simposn’s rule formula becomes:
Then simplifying a bit:
Then we just do those computations and we finally get the approximation via Simposn's rule:
While when we do the integral by using technology we get: 2.3020.
The approximation with Simpon’s rule is close but about 33% smaller: