To solve this problem, let's define the variables and use the Pythagorean Theorem.

Problem Summary:

• There is a point on the ground 35 ft away from the base of a tree.
• The distance from this point to the top of the tree is 1 ft more than three times the height of the tree.
• We need to find the height of the tree.

Step-by-step Solution:

1. Define the Variables:

   • Let $h$ be the height of the tree (in feet).
   • The distance from the point on the ground to the top of the tree is given as $3h + 1$ feet.
   • The distance from the point to the base of the tree (horizontal distance) is 35 ft.

2. Apply the Pythagorean Theorem: Since we have a right triangle formed by the height of the tree, the distance along the ground, and the distance from the point to the top of the tree: $(3h + 1)^2 = h^2 + 35^2$

3. Expand and Simplify the Equation: $(3h + 1)^2 = h^2 + 1225$ $9h^2 + 6h + 1 = h^2 + 1225$ $8h^2 + 6h + 1 - 1225 = 0$ $8h^2 + 6h - 1224 = 0$

4. Solve the Quadratic Equation: Using the quadratic formula: $h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 8$, $b = 6$, and $c = -1224$.

Let's compute the value of $h$.The solutions for the height $h$ of the tree are:

$h_1 = 12.0 \, \text{ft}, \quad h_2 = -12.75 \, \text{ft}$

Since a negative height does not make sense in this context, the height of the tree is:

$\boxed{12 \, \text{ft}}$

The height of the tree is 12 feet when rounded to the nearest foot.

Would you like more details on any part of this solution, or do you have any questions? Here are some related questions you might find interesting:

1. How is the Pythagorean Theorem used in real-life applications?
2. What are the steps to solving a quadratic equation using the quadratic formula?
3. Can you explain the significance of the discriminant in the quadratic formula?
4. How can we use geometry to solve problems involving distances and heights?
5. What are other methods to solve quadratic equations besides the quadratic formula?

Tip: When solving a word problem, clearly define all variables and ensure that each part of the problem is represented in the equation you set up.