A’Leila is building a pen for her pet donkey. The pen is a rectangle with one side measuring b yards and the adjacent side measuring a yards.
A’Leila knows that a=1/3b
a. Write two different expressions giving the perimeter of the donkey’s pen. Using variable “b” only.
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b.Use the value a=12. Evaluate the perimeter of the pen for that value, using one of the expressions that you wrote.
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a. To write the expressions for the perimeter of the donkey's pen using only the variable "b", we first need to express "a" in terms of "b". Since A'Leila knows that a = 1/3b, we can use this relationship to rewrite the perimeter in terms of "b".
The perimeter (P) of a rectangle is given by the formula P = 2(length + width), where in this case, the length is "b" and the width is "a".
Using the given relationship a = 1/3b, let's write two different expressions for the perimeter:
First Expression:
P = 2(b + a)
= 2(b + 1/3b)
= 2b + 2/3b
= 2b + (2/3)b
= (6/3)b + (2/3)b
= (6/3 + 2/3)b
= (8/3)b
Second Expression:
Let's factor out the common factor b from the first expression:
P = 2b + 2/3b
= (2 + 2/3)b
= (6/3 + 2/3)b
= (8/3)b
So you see, factoring doesn't really produce a "different" expression here—mathematically, it's the same expression. However, we can express it differently in terms of order, like so:
Reordered Expression:
P = 2/3b + 2b
= (2/3 + 6/3)b
= (8/3)b
Each of these expressions ultimately simplifies to the same final form, so the perimeter in terms of "b" alone is (8/3)b. This is the simplest form of the expression.
b. When a = 12, we apply this value to the relationship a = 1/3b to find the value of "b":
12 = 1/3b
To solve for b, multiply both sides by 3:
3 * 12 = b
36 = b
So, "b" is 36 yards.
Now, using one of the expressions for the perimeter:
P = (8/3)b
Plug in the value of "b" which is 36 yards:
P = (8/3) * 36
= 8 * (36/3)
= 8 * 12
= 96
Therefore, the perimeter of the pen, when a = 12 and consequently b = 36, is 96 yards.