Chapter 2, Problem 9P

(a)

To determine

To find:The temperature is exactly the same at two opposite points.

Answer to Problem 9P

The temperature is exactly the same at two opposite points is zero as proved.

Explanation of Solution

Given information:

The given function is $S\left(x\right)=T\left(x-180\right)-T\left(x\right)$ .

Calculation:

A point $x$ on the globe and a point that is opposite (which is $180°$ ) $x+180$ , there exist a function $S\left(x\right)=T\left(x-180\right)-T\left(x\right)$ .

Let the temperature is $0°$ .

Then the temperature at $0°$ equals the temperature at $0°+180°$ .

For $S\left(x\right)>0$ .

  $\begin{array}{c}S\left(x+180\right)=T\left(x-180+180\right)-T\left(x+180\right)\\ =T\left(x\right)-T\left(x+180\right)\\ =-S\left(x\right)\end{array}$

This is a opposite function of the original and is also continuous, because temperature is different on a continuous basis.

The value of $S\left(x\right)$ is zero on the interval $\left[x,x+180\right]$ .

For $S\left(x\right)<0$ .

  $\begin{array}{c}S\left(x-180\right)=T\left(x-180-180\right)-T\left(x-180\right)\\ =T\left(x-360\right)-T\left(x-180\right)\\ =T\left(x\right)-T\left(x-180\right)\\ =-S\left(x\right)\end{array}$

This is a opposite function of the original and is also continuous, because temperature is different on a continuous basis.

The value of $S\left(x\right)$ is zero on the interval $\left[x,x-180\right]$ .

Therefore, the temperature is exactly the same at two opposite points is zero as proved.

(b)

To determine

To find:whether the result in part (a)hold for point lying on any circle on the earth’s surface.

Answer to Problem 9P

Yes, the result in part (a) hold for point lying on any circle on the earth’s surface.

Explanation of Solution

Calculation:

Yes, the same argument applies as for part (a) on the point that lies on any circle since temperature varies on a continuous basis at different points on the globe.

Therefore,yes,the result in part (a) hold for point lying on any circle on the earth’s surface.

(c)

To determine

To find:Whether the result in part (a) hold for barometric pressure and for altitude above sea level.

Answer to Problem 9P

Yes, the result in part (a) hold for barometric pressure and for altitude above sea level.

Explanation of Solution

Calculation:

Yes, the same argument applies as foraltitude and barometric pressure since there is always a value for these measurements at a point on the globe.

These values differ on continuous basis, there is a zero for each of these values.

Therefore, the temperature, altitude, and barometric pressure are the same values on the opposite points of the globe.

Therefore,yes, the result in part (a) hold for barometric pressure and for altitude above sea level.