The lapse rate is the rate at which the temperature in Earth’s atmosphere decreases with altitude. For example, a lapse rate of 6.5° Celsius/km means the temperature decreases at a rate of 6.5°C per kilometer of altitude. The lapse rate varies with location and with other variables such as humidity. However, at a given time and location, the lapse rate is often nearly constant in the first 10 kilometers of the atmosphere. A radiosonde (weather balloon) is released from Earth’s surface, and its altitude (measured in km above sea level) at various times (measured in hours) is given in the table below.

a. Assuming a lapse rate of 6.5° C/km, what is the approximate rate of change of the temperature with respect to time as the balloon rises 1.5 hours into the flight? Specify the units of your result, and use a forward difference quotient when estimating the required derivative.

b. How does an increase in lapse rate change your answer in part (a)?

c. Is it necessary to know the actual temperature to carry out the calculation in part (a)? Explain.

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0.5 1 Time (hr) 1.5 2.5 Altitude (km) 0.5 1.2 1.7 2.1 2.5 2.9