Question

if the length of a rectangle is given by the expression (2sqrt{153}) and the width is given by (4sqrt{68}), which radical expression represents the perimeter of the rectangle?
a (6sqrt{34})
b (14sqrt{17})
c (28sqrt{17})
d (32sqrt{17})

Explanation:

Step1: Simplify the length term

Factor $\sqrt{153}$: $\sqrt{153}=\sqrt{9\times17}=3\sqrt{17}$
So length: $2\sqrt{153}=2\times3\sqrt{17}=6\sqrt{17}$

Step2: Simplify the width term

Factor $\sqrt{68}$: $\sqrt{68}=\sqrt{4\times17}=2\sqrt{17}$
So width: $4\sqrt{68}=4\times2\sqrt{17}=8\sqrt{17}$

Step3: Apply perimeter formula

Perimeter of rectangle: $P=2(\text{length}+\text{width})$
Substitute values: $P=2(6\sqrt{17}+8\sqrt{17})$

Step4: Calculate the final value

Add radicals inside parentheses: $6\sqrt{17}+8\sqrt{17}=14\sqrt{17}$
Multiply by 2: $P=2\times14\sqrt{17}=28\sqrt{17}$

Answer:

C. $28\sqrt{17}$