Write, in factored form, an expression for the area of the constructed figure below if A is unknown and B is equal to $6$ . $Write, in factored form, an expression for the area of the constructed figure below if A is unknown and B is equal to 6 . A) A ( A + 6 ) B) ( A - 6 ) ^ { 2 } C) ( A + 6 ) ^ { 2 } D) ( A + 6 ) ( A - 6 ) E) A ( A - 6 )$

A) $A (A + 6)$
B) $A - 6 ) ^ { 2 }$
C) $A + 6 ) ^ { 2 }$
D) $(A + 6) (A - 6)$
E) $A (A - 6)$

Question

Write, in factored form, an expression for the area of the constructed figure below if A is unknown and B is equal to

6

. $Write, in factored form, an expression for the area of the constructed figure below if A is unknown and B is equal to 6 . A) A ( A + 6 ) B) ( A - 6 ) ^ { 2 } C) ( A + 6 ) ^ { 2 } D) ( A + 6 ) ( A - 6 ) E) A ( A - 6 )$

A)

A (A + 6)

B)

A - 6 ) ^ { 2 }

C)

A + 6 ) ^ { 2 }

D)

(A + 6) (A - 6)

E)

A (A - 6)

Rahul Radhesh · Accepted Answer

Final Answer : C, Explanation :   The constructed figure is a rectangle with one side of length A and the other side of length B, which is known to be 6. Therefore, the area of the rectangle is A times 6, or 6A. However, the figure also has two identical quarter-circles with radius B = 6, which add to the rectangle to form the full figure. The area of each quarter-circle is (1/4) times pi times the radius squared, or (1/4) times pi times 6 squared, which simplifies to 9pi. Therefore, the total area of the figure is the area of the rectangle, 6A, plus the combined area of the two quarter-circles, which is 2 times 9pi, or 18pi. Factoring out 6 from 6A gives 6(A), and factoring out 9pi from 18pi gives 9pi(2). Therefore, the total area can be expressed as 6A + 9pi(2), or in factored form as (A + 6)(A + 6), which simplifies to (A + 6)². Therefore, the best choice is C,   ( A + 6 ) 2 ( A + 6 ) ^ { 2 } ( A + 6 ) 2  .

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