Question

The vertices of a quadrilateral lie on the graph of $y = \ln x,$ and the $x$ -coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln\dfrac{91}{90}.$ What is the $x$ -coordinate of the leftmost vertex?

$6$

$7$

$10$

$12$

$13$

Answer: D

Concepts:shoelace formula logarithm quadratic

Difficulty rating: 1860

Solution:

Let the vertices have $x$ -coordinates $n, n+1, n+2, n+3$ with $y$ -coordinates $\ln$ of those values. Applying the shoelace formula and simplifying, the area is $\ln\dfrac{(n+1)(n+2)}{n(n+3)}.$

Setting $\dfrac{(n+1)(n+2)}{n(n+3)} = \dfrac{91}{90}$ gives $1 + \dfrac{2}{n^2 + 3n} = \dfrac{91}{90},$ so $n^2 + 3n = 180.$

Then $(n - 12)(n + 15) = 0,$ so $n = 12.$

Thus, D is the correct answer.

2020 AMC 12A Problem 17

Answer: D

Difficulty rating: 1860

Solution:

Problem 17 in Other Years