Answer:
[tex]Perimeter = \frac{39t-4}{4}[/tex]
Step-by-step explanation:
Given
[tex]Lengths: (2\frac{1}{4}t - 5), (4t + 3), (1/2t - 1), (3t + 2)[/tex]
Required
Determine the perimeter
The perimeter is calculated as the sum of the lengths of the irregular shape.
i.e.
[tex]Perimeter = (2\frac{1}{4}t - 5) + (4t + 3) + (1/2t - 1) + (3t + 2)[/tex]
Remove brackets
[tex]Perimeter = 2\frac{1}{4}t - 5 + 4t + 3 + 1/2t - 1 + 3t + 2[/tex]
Collect Like Terms
[tex]Perimeter = 2\frac{1}{4}t +4t + 1/2t + 3t - 5 + 3 - 1 + 2[/tex]
[tex]Perimeter = 2\frac{1}{4}t +4t + 1/2t + 3t -1[/tex]
Convert mixed number to improper fraction.
[tex]Perimeter = \frac{9}{4}t +4t + 1/2t + 3t -1[/tex]
Take LCM
[tex]Perimeter = \frac{9t + 16t + 2t + 12t}{4} -1[/tex]
[tex]Perimeter = \frac{39t}{4} -1[/tex]
Take LCM
[tex]Perimeter = \frac{39t-4}{4}[/tex] --- The perimeter of the shape
We want to find the perimeter of the irregular quadrilateral given that we know its lengths. By direct computation, we will get:
P(t) = (39/4)*t - 1
Remember that for any figure, the perimeter is defined as the sum of the lengths of each side.
Here we have that the lengths of the four sides of the quadrilateral are:
(2 1/4t - 5)
(4t + 3)
(1/2t - 1)
(3t + 2)
So we must sum that, we will get:
P = (2 1/4t - 5) + (4t + 3) + (1/2t - 1) + (3t + 2)
Notice that in the first part we have a mixed number, we can rewrite it as:
2 + 1/4 = 8/4 + 1/4 = 9/4
Then the sum is:
P = ((9/4)t - 5) + (4t + 3) + (1/2t - 1) + (3t + 2)
Now we simplify this sum:
P = (9/4 + 4 + 1/2 + 3)*t + (-5 + 3 - 1 + 2)
P = (39/4)*t - 1
So the perimeter, as a function of t, is:
P(t) = (39/4)*t - 1
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