Which of the following polynomial equations could be used to represent the area of the trapezoid shown in the figure?

Options

A. `A=k^(2)+k`
B. `A=(3k^(2)+3k)/(2)`
C. `A=(3k^(2)+k)/(2)`
D. `A=(3k^(3)+k^(2))/(2)`

Answer

B

Explanation

Difficult: Hard
Category: Passport to Advanced Math/ Exponents
Strategic advice: The area of a trapezoid is given by the formula `A=(1)/(2)(b_(1)+b_(2))h`,where `b_(1) and b_(2)` represent the lenghts of the bases and h represents the height of the trapezoid.
Getting to the Answer: Although this question asks about a polynomial equation, focus on what you're looking for -an equation that represents the area of the trapezoid. The polynomial part will take care of itself. Examine the figure along with the labels along the axes to determine the dimensions of the trapezoid (which happen to be variables instead of numbers). The height of the trapezoid is the vertical distance from the x-axis to the top base, which is given (labeled on the y-axis) as k+1. The length of the top base is the distance from the y-axis to the doted line inside the trapezoid, which is k. Finally, the length of the bottom base is the distance from the y-axis to the right-most point along th x-axis,which is 2k. Substitute these values into the area formula and simplify:
`A=(1)/(2)(b_(1)+b_(2))h =(1)/(2)(k+2k)(k+1)=(1)/(2)(3k)(k+1)=(1)/(2)(3k^(2)+3k)=(3k^(2)+3k)/(2)`
This matches (B).