Questions: It takes 29 days for the Moon to go through its cycle of full moon, half moon, new moon, half moon, and then back to full moon. The function P models the percentage of the Moon that is visible t days after the cycle begins, where t=0 corresponds to a full moon on the night of June 5, 2020. Based on the model: which of the following best describes how the percentage of the Moon that is visible is changing on the night of June 25, 2020? It is increasing by 8.3% per day. It is increasing by 10.1% per day. It is increasing by 31.5% per day. It is decreasing by 8.3% per day. It is decreasing by 10.1% per day.

It takes 29 days for the Moon to go through its cycle of full moon, half moon, new moon, half moon, and then back to full moon. The function P models the percentage of the Moon that is visible t days after the cycle begins, where t=0 corresponds to a full moon on the night of June 5, 2020. Based on the model: which of the following best describes how the percentage of the Moon that is visible is changing on the night of June 25, 2020?
It is increasing by 8.3% per day.
It is increasing by 10.1% per day.
It is increasing by 31.5% per day.
It is decreasing by 8.3% per day.
It is decreasing by 10.1% per day.

Transcript text: It takes 29 days for the Moon to go through its cycle of full moon, half moon, new moon, half moon, and then back to full moon. The function $P$ models the percentage of the Moon that is visible $t$ days after the cycle begins, where $t=0$ corresponds to a full moon on the night of June 5, 2020. Based on the model: which of the following best describes how the percentage of the Moon that is visible is changing on the night of June 25, 2020? It is increasing by $8.3 \%$ per day. It is increasing by $10.1 \%$ per day. It is increasing by $31.5 \%$ per day. It is decreasing by $8.3 \%$ per day. It is decreasing by $10.1 \%$ per day.

Solution

Solution Steps

Step 1: Define the Function

The function modeling the percentage of the Moon that is visible $ P(t) $ is given by: \[ P(t) = 50 \cos \left(\frac{2 \pi}{29} t\right) + 50 \] where $ t $ is the number of days after the cycle begins, with $ t = 0 $ corresponding to a full moon.

Step 2: Calculate the Derivative

To find how the visibility of the Moon is changing, we calculate the derivative $ P'(t) $: \[ P'(t) = -50 \cdot \frac{2 \pi}{29} \sin \left(\frac{2 \pi}{29} t\right) \]

Step 3: Evaluate the Derivative at $ t = 20 $

We evaluate the derivative at $ t = 20 $ days (June 25, 2020): \[ P'(20) \approx -50 \cdot \frac{2 \pi}{29} \sin \left(\frac{2 \pi}{29} \cdot 20\right) \] Calculating this gives us: \[ P'(20) \approx 10.1\% \]

Final Answer

The percentage of the Moon that is visible is increasing by approximately $ 10.1\% $ per day on the night of June 25, 2020. Thus, the answer is: \[ \boxed{10.1\%} \]

Was this solution helpful?

Unhelpful

Helpful

Questions asked by other users

Next >

Thank you for your feedback.

Copied!