Let's solve the problem step by step:

Problem Breakdown:

1. Jimmy is halfway between two trees that are 200 feet apart. Therefore, Jimmy is 100 feet from each tree.
2. The angle of elevation to the top of the pine tree is $37^\circ$.
3. The angle of elevation to the top of the spruce tree is $26^\circ$.
4. We need to calculate:
   • The height of both trees using trigonometry.
   • Which tree is taller and by how much.

Step 1: Set up Trigonometric Relationships

The height ($h$) of a tree can be calculated using the tangent function: $\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}$ Where:

• Opposite = height of the tree ($h$)
• Adjacent = distance between Jimmy and the tree (100 feet for both trees)

Rearranging the formula: $h = \tan(\text{angle}) \cdot \text{adjacent}$

Step 2: Calculate the Heights

For the Pine Tree ($37^\circ$):

$h_\text{pine} = \tan(37^\circ) \cdot 100$ Using a calculator: $\tan(37^\circ) \approx 0.7536$ $h_\text{pine} = 0.7536 \cdot 100 = 75.36 \, \text{feet}$

For the Spruce Tree ($26^\circ$):

$h_\text{spruce} = \tan(26^\circ) \cdot 100$ Using a calculator: $\tan(26^\circ) \approx 0.4877$ $h_\text{spruce} = 0.4877 \cdot 100 = 48.77 \, \text{feet}$

Step 3: Compare Heights

• Height of Pine Tree: $75.36 \, \text{feet}$
• Height of Spruce Tree: $48.77 \, \text{feet}$

The pine tree is taller by: $75.36 - 48.77 = 26.59 \, \text{feet}$

Final Answer:

• The pine tree is taller.
• It is taller by 26.59 feet.

Would you like a detailed explanation of any step? Here are some follow-up questions:

1. How does the tangent function work, and why is it used here?
2. Can you show the calculations for a different angle or distance?
3. What happens if Jimmy isn't halfway between the trees?
4. Can we calculate the angle if we only know the height and distance?
5. How can trigonometry be applied in real-world situations like this?

Tip: Always sketch a diagram for problems involving angles and distances. It helps visualize the problem better!