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) $A ( A - 6 )$

Question

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

Quizplus · Accepted Answer

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, $( A + 6 ) ^ { 2 }$.

Write, in Factored Form, an Expression for the Area of the Constructed