The National Weather Service releases approximately

Question

The National Weather Service releases approximately $70,000$ radiosondes every year to collect data from the atmosphere. Attached to a balloon, a radiosonde rises at about $1000$ ft/min until the balloon bursts in the upper atmosphere. Suppose a radiosonde is released from a point $6$ ft above the ground and that $5$ seconds later, it is $83$ ft above the ground. Let $f\left(t\right)$ represent the height (in feet) that the radiosonde is above the ground $t$ seconds after it is released. Evaluate $\frac{f\left(5\right)-f\left(0\right)}{5-0}$ and interpret the meaning of this quotient.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem dealing with rates. This problem says a hiker starts climbing a hill from an elevation of 200 ft above sea level. After 20 minutes, the hiker reaches an elevation of 800 ft above sea level. LH, as a function of t represent the elevation in feet above sea level of the hiker minutes. After starting the climb, we need to evaluate the expression, the quantity of H of 20 minus H of zero in quantity divided by the quantity of 20 minus zero and select the correct interpretation of this quotient. We're given four possible choices as our answers. For choice. A we have 30 ft per minute. The hiker climbs at an average rate of 30 ft per minute during the last 20 minutes of the hike. For Choice B, we have 6 ft per minute. The hiker climbs at an average rate of 6 ft per minute during the 1st 20 minutes of the hike. For Choice C we have 30 ft per minute. The hiker climbs an average rate of 30 ft per minute during the 1st 20 minutes of the hike. And for Choice D we have 20 ft per minute. The hiker climbs at an average rate of 20 ft per minute during the 1st 20 minutes of the hike. Now, this question asks us to do two things, evaluate an expression and to interpret that expression. So we're gonna start off with the evaluation and the expression that we need to evaluate is the quantity of each of 20 minus H of zero in quantity divided by the quantity of 20 minus zero. Now, we're told that the function H of T represents the elevation in feet above sea level. And so we need to determine what those values are for the two functions that we are given the age of zero and the age of 20. So we'll start off with H of zero and this is gonna be equal to our initial elevation. And we're told that the hiker starts to climb at an elevation of 200 ft above sea level, sea levels. That means of zero is going to be equal to 200 ft and that occurs at T equal to zero minutes. That's our starting point. Likewise, we need to evaluate age of 20 and that is going to be equal to 800 ft above sea level because we're told that after 20 minutes, the hiker reaches an elevation of 800 ft. So age of 20 is gonna be equal to 800 feet and that occurs at T equal to 20 minutes. So now, we can plug those values into our expression that we need to evaluate. So our quantity of H of 20 minus H of zero in quantity divided by the quantity of 20 minus zero is going to be equal to or of 20. That is going to be the 800 ft. So I'll have 800 minus are H zero, which is the 200 ft. So minus the 200 that's gonna be divided by the quantity of 20 minus zero. So I can evaluate the expression on the right hand side. So I'll have each of 20 minus H of zero. And that quantity is gonna be divided by the quantity of 20 minus zero, gonna be equal to 30. And here this is going to have units of feet per minute. That's because our height, we, we're in feet, which is what's in the numerator and our denominator there is the time which has units of minutes. And so this is what the question is going to evaluate too. So that's half of our problem answered. The other half is that we need to correctly interpret what this means. And so, since I'm taking this, um, this rate over a long time period of 20 minutes, that means that this is going to be an average rate. So this is the average rate at which the hiker is climbing. And this actually occurs during the 1st 20 minutes since my initial um, height is taken at t equal to zero minutes. And our final height is taken after 20 minutes. So this climb, this uh rate is actually measured during the 1st 20 minutes of the hike. So now if I look at my answer choices, this the correct answer choice corresponds to answer choice. C since we have 30 ft per minute as the evaluation for the expression, and that the hiker climbs an average rate of 30 ft per minute during the 1st 20 minutes of the hike. So I hope that this explanation has been useful and I'll see you in the next video.

Key Concepts

Function Evaluation

Average Rate of Change

Interpretation of Results

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