To solve this problem, we need to determine the maximum height you could be while still being completely within the shadow of the tree. This can be approached using similar triangles.

Step-by-Step Solution

1. Understand the Problem Setup:

   • A 21-foot tall tree casts a 35-foot shadow.
   • You are standing 30 feet away from the tree.
   • We need to find out the maximum height you can be while remaining completely in the shadow of the tree.

2. Identify Similar Triangles:

   • The shadow of the tree and the height of the tree form a right triangle.
   • Similarly, your height and the distance from you to the point where the shadow ends also form a right triangle.
   • Since the sun's rays are the same for both triangles, the triangles are similar.

3. Set Up the Proportion Using Similar Triangles:

   Let's define:

   • $h$ as your height.
   • The total length from the tree to the end of the shadow is 35 feet.
   • Since you are standing 30 feet away from the tree, the distance from you to the end of the shadow is $35 - 30 = 5$ feet.

   The two similar triangles are:

   • Triangle 1 (Tree's shadow): Height = 21 feet, Shadow = 35 feet.
   • Triangle 2 (Your shadow): Height = $h$, Shadow = 5 feet.

   The ratios of the corresponding sides in similar triangles are equal: $\frac{\text{Height of the tree}}{\text{Shadow length of the tree}} = \frac{\text{Your height}}{\text{Your shadow length}}$

   Substituting the known values: $\frac{21}{35} = \frac{h}{5}$

4. Solve for Your Height $h$:

   First, simplify the fraction on the left: $\frac{21}{35} = \frac{3}{5}$

   Now set up the equation: $\frac{3}{5} = \frac{h}{5}$

   Cross multiply to solve for $h$: $3 \times 5 = 5 \times h$ $15 = 5h$

   Divide both sides by 5: $h = 3$

Conclusion

You could be 3 feet tall and still be completely within the shadow of the 21-foot tree.

Would you like more details or have any questions?

Related Questions to Explore:

1. What would be your height if the distance between you and the tree changed?
2. How would the solution change if the tree's height increased to 25 feet?
3. If you were 6 feet tall, how far would you have to stand from the tree to remain in its shadow?
4. How long would the shadow be if the tree were 30 feet tall instead of 21 feet?
5. If the sun moved, shortening the shadow to 25 feet, what would be your maximum height to stay within the shadow?

Tip: Understanding similar triangles is crucial for solving many real-world problems involving shadows and perspectives.