### Home > APCALC > Chapter 5 > Lesson 5.2.1 > Problem5-57

5-57.

A weather balloon is launched and the following data is sent to the meteorological center. The column marked time, $t$, is given in seconds after launch, elevation, $e$, is in feet above sea level, and temperature, $T$, is in degrees Fahrenheit.

 Time ($t$) Elevation ($e$) Temp ($T$) $0$ $1260$ $56.4^\circ$ $30$ $1560$ $55.1^\circ$ $60$ $1920$ $53.9^\circ$ $90$ $2350$ $51.9^\circ$ $120$ $2750$ $49.5^\circ$ $150$ $3170$ $47.7^\circ$ $180$ $3600$ $45.0^\circ$ $210$ $4080$ $42.4^\circ$ $240$ $4560$ $40.9^\circ$

Since the derivative (IROC) cannot be determined, calculate the average rate of change.
For $e\left(t\right)$:

1. Approximately how fast is the balloon rising at $120$ seconds?

$\text{You can use Hana's method: }\frac{e(150)-e(120)}{150-120}$

$\text{or Anah's method: }\frac{e(120)-e(90)}{120-90}$

$\text{or Hanah's method: }\frac{e(150)-e(90)}{150-90}$

2. Approximately how fast is the temperature changing at $2750$ feet elevation?

Use Hana, Anah, or Hanah's method to approximate the rate of change for $T\left(e\right)$ at $e = 2750$.

3. Approximately how fast is the temperature changing at $120$ seconds?

Use Hana, Anah, or Hanah's method to approximate the rate of change for $T\left(t\right)$ at $t = 120$.

4. Why would you expect the product of the answers to (a) and (b) to equal answer (c)?

$\text{Rewrite: }e'(t)=\frac{de}{dt} \text{ and }T'(e)=\frac{dT}{de}$

$\frac{de}{dt}\cdot \frac{dT}{de}=$