Question

The National Weather Service releases approximately ﻿70,00070,00070,000﻿ radiosondes every year to collect data from the atmosphere. Attached to a balloon, a radiosonde rises at about ﻿100010001000﻿ ft/min until the balloon bursts in the upper atmosphere. Suppose a radiosonde is released from a point ﻿666﻿ ft above the ground and that ﻿555﻿ seconds later, it is ﻿838383﻿ ft above the ground. Let ﻿f(t)f\left(t\right)f(t)﻿ represent the height (in feet) that the radiosonde is above the ground ﻿ttt﻿ seconds after it is released. Evaluate ﻿f(5)−f(0)5−0\frac{f\left(5\right)-f\left(0\right)}{5-0}5−0f(5)−f(0)​﻿ and interpret the meaning of this quotient.

Accepted Answer

Identify the given information: The radiosonde is released from a height of 666 ft and reaches a height of 8383 ft after 555 seconds. Recognize that the function f(t) represents the height of the radiosonde at time t. We need to evaluate the average rate of change of the height over the first 5 seconds, which is given by the expression $$ \frac{f(5) - f(0)}{5 - 0} . $$Understand that $$ f(5) $$ represents the height of the radiosonde 5 seconds after release, and $$ f(0)  is $$the initial height, which is 666 ft. The expression $$ \frac{f(5) - f(0)}{5} $$ calculates the average rate of change of the radiosonde's height over the first 5 seconds. This is essentially the average velocity of the radiosonde during this time interval. Interpret the meaning: The quotient represents the average rate at which the radiosonde's height is increasing per second over the first 5 seconds after its release.

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Key Concepts

Function Evaluation

Average Rate of Change

Interpretation of Results