The length of a rectangle is represented by the function l(x) = 5x. the width of that same rectangle is represented by the function w(x) = 2x^2 − 4x + 13. which of the following shows the area of the rectangle in terms of x?

Remember that the area of a rectangle is the length of the rectangle multiplied by the width of the rectangle.

In this case, we could say (where is the area of the rectangle):

Substituting the values the problem gave us for and , we can find the formula for in terms of , which is:

The formula for the area of the rectangle would be A(x) = 10x³ - 20x² + 65x.

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!