The problem involves two hot air balloons seen at different angles from the vertical. To determine how much higher the right balloon is than the left, we can apply trigonometry.

Step-by-step approach:

1. Lower Balloon:

   • Distance from Josh: 700 meters.
   • Angle from vertical: $38^\circ$.
   • To find the height of this balloon from the ground, we use the cosine function: $h_{\text{lower}} = 700 \times \cos(38^\circ)$

2. Higher Balloon:

   • Distance from Josh: 1050 meters.
   • Angle from vertical: $26^\circ$.
   • To find the height of this balloon from the ground: $h_{\text{higher}} = 1050 \times \cos(26^\circ)$

3. Difference in Height:

   • The difference in height between the two balloons is given by: $\Delta h = h_{\text{higher}} - h_{\text{lower}}$

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In the meantime, if you'd like to manually calculate the difference in height:

1. Use the formula $\text{height} = \text{distance} \times \cos(\text{angle})$.
2. For the lower balloon, use 700 meters and an angle of 38°.
3. For the higher balloon, use 1050 meters and an angle of 26°.
4. Subtract the two heights to find the difference.

Would you like further details or clarification on how to do this? Here are five related questions you can explore:

1. How do trigonometric functions relate to vertical and horizontal distances?
2. How would the problem change if the angles were measured from the horizontal instead?
3. What if both balloons were moving upward? How would their speed affect the calculations?
4. How can we calculate the distance between the two balloons in 3D space?
5. How does this type of problem relate to real-world navigation and surveying?

Tip: Always ensure your calculator is set to degrees (not radians) when working with degree measurements.