Chapter 1.PPS, Problem 25P

To determine

To show:

a) At any fixed time there are at least two diametrically opposite point on the equator that have exactly the same temperature.

To find:

b) If part a) holds for point lying on any circle on the earth’s surface.

c) If part a) holds for barometric pressure and for altitude above sea level.

a)

Explanation:

1) Concept: Intermediate value theorem

The intermediate value theorem states that if a continuous function, $f$ with an interval, [a, b], as its domain, takes values $f\left(a\right)$ and $f\left(b\right)$ at each end of the interval, then it also takes any value between $f\left(a\right)$ and $f\left(b\right)$ at some point within the interval.

2) Given:

Start form $0^o$ latitude and proceed in a westerly direction.

 $T\left(x\right)$ denote the temperature at the point $x$ at any given time.

 $T$ is continuous function of $x.$

3) Calculations:

The latitude $x+{180}^o$ and $x$ is diametrically opposite point on the equator.

Let us define $H\left(x\right)=T\left(x+{180}^o\right)-T\left(x\right)$

Take any number $a$

Case 1:

If $H\left(a\right)=0$, because (temperature at $a$) = (temperature at $a+{180}^o$).

Case 2:

If $H\left(a\right)>0$ then

 $H\left(a\right)=T\left(a+{360}^o\right)-T\left(a+{180}^o\right)$

 $H\left(a\right)=T\left(a\right)-T\left(a+{180}^o\right)=-H\left(a\right)<0$

Since here $H$ is continuous then temperature is continuous too.

So by intermediate value Theorem

 $H$ is continuous on the interval $\left[a,a+{180}^o\right]$

Hence two opposite point have same temperature.

b)

Explanation:

1) Concept: Intermediate value theorem

The intermediate value theorem states that if a continuous function, $f$ with an interval, [a, b], as its domain, takes values $f\left(a\right)$ and $f\left(b\right)$ at each end of the interval, then it also takes any value between $f\left(a\right)$ and $f\left(b\right)$ at some point within the interval

2) Given:

Start form $0^o$ latitude and proceed in a westerly direction.

 $T\left(x\right)$ denote the temperature at the point $x$ at any given time.

 $T$ is continuous function of $x.$

3) Calculations:

Suppose the point lying on any circle on the earth’s surface.

The points are the latitude $x+{180}^o$ and $x$ is diametrically opposite point on the circle.

Similarly solve by part a)

c)

Explanation:

1) Concept: Intermediate value theorem

The intermediate value theorem states that if a continuous function, $f$ with an interval, [a, b], as its domain, takes values $f\left(a\right)$ and $f\left(b\right)$ at each end of the interval, then it also takes any value between $f\left(a\right)$ and $f\left(b\right)$ at some point within the interval

2) Given:

Start form $0^o$ latitude and proceed in a westerly direction.

 $T\left(x\right)$ denote the temperature at the point $x$ at any given time.

 $T$ is continuous function of $x.$

3) Calculations:

Barometric pressure fits the bill.

Altitude over the (vertical cliff) may be discontinues there for argument are not always hold.

Conclusion:

At any fixed time there are at least two diametrically opposite points on the equator that have exactly the same temperature. It also holds for barometric pressure.