   Chapter 14, Problem 15PS

Chapter
Section
Textbook Problem

Surface Area Use the result of Exercise 14 to order the planes in ascending order of their surface areas for a fixed region R in the xy-plane. Explain your ordering without doing any calculations.(a) z 1 = 2 + x (b) z 2 = 5 (c) z 3 = 10 − 5 x + 9 y (d) z 3 = 3 + x − 2 y

To determine
The order of planes in ascending order of their surface area by using the result of exercise 14 without doing any calculations. The planes are given below:

(a) z1=2+x

(b) z2=5

(c) z3=105x+9y

(d) z4=3+x2y

Explanation

Given:

The provided planes are:

(a) z1=2+x

(b) z2=5

(c) z3=105x+9y

(d) z4=3+x2y

The result of exercise 14 states that the area of a rectangular region in xy-plane is given as,

A=secθΔxΔy

Explanation:

It can be seen that area is directly proportional to the angle between the provided plane and the xy-plane according to the result of exercise 14, i.e., A=secθΔxΔy,. Hence, the greater the surface area the greater the the angle between the provided plane and the xy-plane

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