To calculate the distance of the moon from Earth, we can use the formula for the angular separation between two points on the Earth's surface and the angle subtended by the moon.
Given:
• The angle subtended at the moon by two directions of observation is θ=1∘54′\theta = 1^\circ 54'.
• The diameter of the Earth D=1.276×107 mD = 1.276 \times 10^7 \, \text{m}.
• The radius of the Earth R=D2=1.276×1072 m=6.38×106 mR = \frac{D}{2} = \frac{1.276 \times 10^7}{2} \, \text{m} = 6.38 \times 10^6 \, \text{m}.
Approach:
The angle subtended by the moon at the Earth can be treated as an angular separation, and we use the formula for angular separation in a circle:
θ=dr\theta = \frac{d}{r}
Where:
• θ\theta is the angle in radians,
• dd is the linear separation (diameter of the Earth),
• rr is the distance to the moon.
However, since the angle is given in degrees, we first need to convert it into radians.
Step 1: Convert the angle into radians
We know that:
• 1∘=π180 radians1^\circ = \frac{\pi}{180} \, \text{radians},
• 54′=5460∘=0.9∘54' = \frac{54}{60}^\circ = 0.9^\circ.
So, the total angle in degrees is:
θ=1∘54′=1+0.9=1.9∘.\theta = 1^\circ 54' = 1 + 0.9 = 1.9^\circ.
Now, convert this into radians:
θ=1.9×π180 radians.\theta = 1.9 \times \frac{\pi}{180} \, \text{radians}. θ=1.9π180=π94.7368 radians.\theta = \frac{1.9 \pi}{180} = \frac{\pi}{94.7368} \, \text{radians}. θ≈0.03351 radians.\theta \approx 0.03351 \, \text{radians}.
Step 2: Use the formula to calculate the distance to the moon
Now, use the formula for angular separation:
θ=dr,\theta = \frac{d}{r},
where d=6.38×106 md = 6.38 \times 10^6 \, \text{m} is the diameter of the Earth, and rr is the distance to the moon.
Rearranging to solve for rr:
r=dθ=6.38×1060.03351.r = \frac{d}{\theta} = \frac{6.38 \times 10^6}{0.03351}.
Step 3: Perform the calculation
r≈1.9×108 m.r \approx 1.9 \times 10^8 \, \text{m}.
Final Answer:
The distance of the moon from Earth is approximately 1.9 × 10^8 meters, or 190,000 km.
