Questions: Alonzo works for NASA designing planetary rovers. For his current project, he has to design a rover capable of withstanding the harsh climate on the surface of Venus. As part of his design work, Alonzo models the daily high surface temperature of Venus using a normal distribution with a mean of 468°C and a standard deviation of 30°C. Use this table or the ALEKS calculator to find the percentage of daily high temperatures between 444°C and 486°C according to the model. For your intermediate computations, use four or more decimal places. Give your final answer to two decimal places (for example 98.23%).

Solution Steps

The daily high surface temperature of Venus is modeled using a normal distribution with the following parameters:

• Mean (\( \mu \)): \( 468 \, ^\circ \mathrm{C} \)
• Standard Deviation (\( \sigma \)): \( 30 \, ^\circ \mathrm{C} \)

To find the probability of daily high temperatures between \( 444 \, ^\circ \mathrm{C} \) and \( 486 \, ^\circ \mathrm{C} \), we first calculate the Z-scores for the bounds:

1. For \( 444 \, ^\circ \mathrm{C} \): \[ Z_{start} = \frac{444 - 468}{30} = \frac{-24}{30} = -0.8 \]

2. For \( 486 \, ^\circ \mathrm{C} \): \[ Z_{end} = \frac{486 - 468}{30} = \frac{18}{30} = 0.6 \]

Using the Z-scores, we can find the probability that the daily high temperatures fall within the specified range: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(0.6) - \Phi(-0.8) \] From the standard normal distribution table, we find:

• \( \Phi(0.6) \approx 0.7257 \)
• \( \Phi(-0.8) \approx 0.2119 \)

Thus, the probability is: \[ P \approx 0.7257 - 0.2119 = 0.5138 \]

To express the probability as a percentage: \[ P \times 100 \approx 51.38\% \]

Final Answer