Medical Case Study on Gastronomic System: Symptoms, Testing, and Diagnosis

Part A – The Anatomy

Dr. Andersen, a specialist on the human gastronomic system, determined that many of the symptoms elicited by Mathews could be directly related to a problem in either the small or large intestine.  A battery of tests were performed on Mathews, two noteworthy results are described below. 

The first procedure was performed in the interest of collecting bacterial culture swabs of Mathews’ small intestine.  A long flexible tube is passed through the nose, down the throat and esophagus and through the stomach.  A small camera, attached to the top of the tube recorded every twist and turn of the journey. It was performed under x-ray guidance.  The data from both the camera and the x-ray machine were used to create a detailed sketch of Mathews’ gastronomic tract. 

Question 1  (10 marks)

 A specific section of Mathews’ gastronomic tract can be modeled by the function, where x represents distance traveled by the scope, in cm, and refers to the vertical height within the body relative to the belly button, in cm.

a)(5 marks) Rewrite this equation in factored form.  Show all of your work.

b)(2 marks) Use this information to sketch a graph, by hand, of this section of Mathews’ small intestine.

c)(1 mark) Determine the domain of this function.

d)(2 marks) Bacterial culture samples were taken at two unique points along the journey. Clearly mark these points on your graph.

At the first turning point
At the only root with order two

The second procedure was performed in the interest of collecting a biopsy of a section of the small intestine. The biopsy was removed surgically and examined quantitatively and qualitatively.  Faint, yet distinct, areas of light and dark tissue were discovered on the surface of the small sample.

Question 2    (10 marks)

Assume that a 20mm by 20mm section of tissue was placed on top of a Cartesian plane with its center at the origin.  The darker, shaded areas found on the tissue sample satisfy the rational inequality.

a)(8 marks) Sketch a graph of the function  by hand.

b)(2 marks) Shade the region(s) of the graph that satisfy the inequality to reveal the light and dark regions of the biopsy noted by Dr. Andersen.

With the help of Mathews, Dr. Andersen constructed a timeline of the last 18 months.  They mapped regions based on the intensity of his symptoms and found that there was a cyclical pattern. 

Question 3   (6 marks)

The cyclical pattern of symptoms is sinusoidal in nature.  When graphed as a function of intensity, , and time, x  in months, the function was very similar to
 

Dr. Andersen described shape of this graph, with reference to .

Intensity is ranked on a scale of 0-10, where zero represents normal and ten represents full blown symptoms.  The amplitude of the cyclical pattern is 5 and the equation of the axis of the curve is.  The period of the function is 4 months.  Using  as a base function, the symptoms function has been horizontally translated 2 months to the right.

a)(4 marks) Write a possible equation for the function .

b)(2 marks) Sketch a graph of the function, by hand, on the interval .