Let's go through this problem step by step.
### Part (a): Find the length of the rectangle
The area of the rectangle is given by:
[tex]\[ \text{Area} = 12a^2 + 16ab \][/tex]
The breadth of the rectangle is given by:
[tex]\[ \text{Breadth} = 4a \][/tex]
The area of a rectangle is also given by the product of its length and breadth. Therefore:
[tex]\[ \text{Area} = \text{Length} \times \text{Breadth} \][/tex]
Using the given expressions:
[tex]\[ 12a^2 + 16ab = \text{Length} \times 4a \][/tex]
To find the length, we divide both sides of the equation by [tex]\( 4a \)[/tex]:
[tex]\[ \text{Length} = \frac{12a^2 + 16ab}{4a} \][/tex]
Simplifying the right-hand side:
[tex]\[ \text{Length} = \frac{12a^2}{4a} + \frac{16ab}{4a} \][/tex]
[tex]\[ \text{Length} = 3a + 4b \][/tex]
So the length of the rectangle is:
[tex]\[ \text{Length} = 3a + 4b \][/tex]
### Part (b): Write down the expression for its perimeter
The perimeter (P) of a rectangle is given by the formula:
[tex]\[ P = 2(\text{Length} + \text{Breadth}) \][/tex]
Using the expressions we have for length and breadth:
[tex]\[ \text{Length} = 3a + 4b \][/tex]
[tex]\[ \text{Breadth} = 4a \][/tex]
So, the perimeter is:
[tex]\[ P = 2((3a + 4b) + 4a) \][/tex]
[tex]\[ P = 2(3a + 4b + 4a) \][/tex]
[tex]\[ P = 2(7a + 4b) \][/tex]
Thus, the expression for the perimeter is:
[tex]\[ P = 2(7a + 4b) \][/tex]
### Part (c): Find its actual perimeter if [tex]\( a = 2 \)[/tex] cm and [tex]\( b = 4 \)[/tex] cm
We substitute [tex]\( a = 2 \)[/tex] cm and [tex]\( b = 4 \)[/tex] cm into the perimeter expression:
[tex]\[ P = 2(7a + 4b) \][/tex]
Substituting the values:
[tex]\[ P = 2(7(2) + 4(4)) \][/tex]
[tex]\[ P = 2(14 + 16) \][/tex]
[tex]\[ P = 2(30) \][/tex]
[tex]\[ P = 60 \, \text{cm} \][/tex]
So, the actual perimeter of the rectangle is:
[tex]\[ 60 \, \text{cm} \][/tex]
