Question

Scientists plan to release a space probe that will enter the atmosphere of a gaseous planet. The temperature of the gaseous planet increases linearly with the height of the atmosphere as measured from the top of a visible boundary layer, defined as 0 kilometers in altitude. The instruments on board can withstand a temperature of 601 K. At what altitude will the probe's instruments fail? A. 50 kilometers B. 80 kilometers C. 83 kilometers D. 100 kilometers E. 111 kilometers

Answer 1

Without a specified rate of temperature change with altitude, it is impossible to calculate at which altitude the probe's instruments will fail at 601 K.

The question revolves around determining at what altitude the space probe's instruments will fail when entering a gaseous planet's atmosphere where the temperature increases linearly with altitude. Despite the additional context regarding space probes resisting high temperatures and details on Galileo's mission to Jupiter, those specifics aren't directly useful for answering the question with the given options A to E.

The provided temperature data of Venus' atmosphere (Figure 10.12), which shows temperatures at various altitudes, cannot be directly applied here either since the question outlines a linear increase in temperature, no specific temperatures at known altitudes, or a defined rate of temperature change per kilometer of altitude. Consequently, there isn't enough information to determine at which altitude the instruments will fail at 601 K.

Answer 2

To determine the altitude at which the probe's instruments will fail, we need to find the point where the temperature reaches 601 K.

Let \( T \) be the temperature, \( H \) be the altitude, and \( m \) be the rate of increase.

Since the temperature increases linearly, we can express this relationship as:

\[ T = mH \]

Given that \( T = 601 \) K at the point of failure, we have:

\[ 601 = mH \]

Now, we need to determine the value of \( m \). To find \( m \), we can use the fact that the temperature is given to increase linearly. In the given answer choices, \( m \) is the slope of this linear relationship.

If we consider point A (altitude 0) and point E (altitude 111 km), we can calculate the slope (\( m \)):

\[ m = \frac{\text{change in temperature}}{\text{change in altitude}} \]

\[ m = \frac{601 - 0}{111 - 0} \]

\[ m = \frac{601}{111} \]

Now, we can use \( m \) to find the altitude where the temperature reaches 601 K:

\[ 601 = \frac{601}{111} \times H \]

Solving for \( H \):

\[ H = 111 \]

Therefore, the probe's instruments will fail at an altitude of 111 kilometers. So, the correct answer is E.