Physics quiz question: area factor increase

Physics 205A Quiz 1, Fall Semester 2009
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problems 1.11, 1.12

[Version 1]

An architect is redesigning a rectangular doorway. She decides to double the width of the doorway, and increase the height by 50%. By what factor has the area of the doorway increased?
(A) 1.0
(B) 1.5.
(C) 2.5.
(D) 3.

Correct answer: (D)

The original and new areas A in terms of the original and new widths w and heights h of the doorway are given by:

A_1 = w_1*h_1,
A_2 = w_2*h_2.

The new width and new height, in terms of the original width and height are given by:

w_2 = 2.0*w_1 ("doubled"),
h_2 = 1.50*h_1 ("increased by 50%).

Thus the new area, in terms of the original area is:

A_2 = h_2*w_2 = (2.0*w_1)*(1.50*h_1) = 3.0*w_1*h_1 = 3.0*A_1,

corresponding to an increase by a factor of 3.0.

Response (A) is 2.0*0.50 (the result of which, strangely enough, would correspond to a new doorway that has the same area as the original doorway, despite the increases in width and height). Response (B) is the result of 2.0 - 0.50; and response (C) is 2.0 + 0.50.

Student responses
Sections 70854, 70855
(A) : 5 students
(B) : 7 students
(C) : 11 students
(D) : 32 students

Success level: 58%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.65

[Version 2]

An architect is redesigning a rectangular doorway. She decides to increase the width of the doorway by 40%, and increase the height by 25%. By what factor has the area of the doorway increased?
(A) 0.100.
(B) 1.10.
(C) 1.70.
(D) 1.80.

Correct answer: (D)

The new width and new height, in terms of the original width and height are given by:

w_2 = 1.40*w_1 ("increased by 40%"),
h_2 = 1.25*h_1 ("increased by 25%).

Thus the new area, in terms of the original area is:

A_2 = h_2*w_2 = (1.40*w_1)*(1.25*h_1) = 1.80*w_1*h_1 = 1.80*A_1,

corresponding to an increase by a factor of 1.80.

Response (A) is the result of 0.40*0.25; response (B) is (1+ 0.40*0.25); and response (C) is (1+ 0.40 + 0.25).

Student responses
Section 72177
(A) : 3 students
(B) : 1 student
(C) : 2 students
(D) : 9 students

Success level: 60%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.37