Problem:
The vertices of a quadrilateral lie on the graph of y=lnx, and the x-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is ln9091. What is the x-coordinate of the leftmost vertex?
Answer Choices:
A. 6
B. 7
C. 10
D. 12
E. 13 Solution:
Let the coordinates of the quadrilateral be (n,ln(n)),(n+1,ln(n+1)),(n+2,ln(n+2)),(n+3,ln(n+3)). We have by shoelace's theorem, that the area is
We know that the numerator must have a factor of 13 , so given the answer choices, n is either 12or11 . If n=11, the expression n(n+3)(n+1)(n+2) does not evaluate to 9091, but if n=12, the expression evaluates to 9091. Hence, our answer is (D)12 .
OR
Like above, use the shoelace formula to find that the area of the quadrilateral is equal to lnn(n+3)(n+1)(n+2). Because the final area we are looking for is ln9091, the numerator factors into 13 and 7 , which one of n+1 and n+2 has to be a multiple of 13 and the other has to be a multiple of 7 . Clearly, the only choice for that is (D)12 .