The weight of an object on or beneath the surface of the moon varies directly as its distance from the center of the moon, assuming that the moon has uniform density . The radius of the moon is approximately 1,080 miles. If an object weighs 60 pounds on the surface of the moon, how far beneath the surface, in miles , would it have to be to weigh 50 pounds ?

Answer

To solve the problem, we will follow the steps outlined in the video transcript. Here’s the step-by-step solution:

### Step 1: Understand the relationship between weight and distance
The weight \( W \) of an object on or beneath the surface of the Moon varies directly as its distance \( D \) from the center of the Moon. This can be expressed mathematically as:
\[ W \propto D \]
This means that we can write:
\[ \frac{W_1}{W_2} = \frac{D_1}{D_2} \]
where:
- \( W_1 \) is the weight on the surface (60 pounds),
- \( W_2 \) is the weight we want to find (50 pounds),
- \( D_1 \) is the distance from the center of the Moon when the object is on the surface (radius of the Moon = 1080 miles),
- \( D_2 \) is the distance from the center of the Moon when the object weighs 50 pounds (unknown).

### Step 2: Assign known values
From the problem:
- \( W_1 = 60 \) pounds
- \( W_2 = 50 \) pounds
- \( D_1 = 1080 \) miles

### Step 3: Set up the proportion
Using the relationship established:
\[ \frac{60}{50} = \frac{1080}{D_2} \]

### Step 4: Cross-multiply to solve for \( D_2 \)
Cross-multiplying gives:
\[ 60 \cdot D_2 = 50 \cdot 1080 \]
Now, calculate \( 50 \cdot 1080 \):
\[ 50 \cdot 1080 = 54000 \]
So we have:
\[ 60 \cdot D_2 = 54000 \]

### Step 5: Solve for \( D_2 \)
Now, divide both sides by 60:
\[ D_2 = \frac{54000}{60} = 900 \text{ miles} \]

### Step 6: Find the distance beneath the surface
The distance beneath the surface of the Moon is given by the difference between the radius of the Moon and \( D_2 \):
\[ \text{Distance beneath the surface} = D_1 - D_2 = 1080 - 900 = 180 \text{ miles} \]

### Final Answer
The object must be **180 miles** beneath the surface of the Moon to weigh 50 pounds.

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