Questions: At a point on the ground 15 ft from the base of a tree, the distance to the top of the tree is 1 ft more than 2 times the height of the tree. Find the height of the tree. The height of the tree is ft . (Simplify your answer. Round to the nearest foot as needed.)

Solution Steps

To find the height of the tree, we can use the Pythagorean theorem. Let \( h \) be the height of the tree. The distance from the point on the ground to the top of the tree is given as \( 2h + 1 \). The distance from the point on the ground to the base of the tree is 15 ft. Using the Pythagorean theorem, we can set up the equation:

\[ 15^2 + h^2 = (2h + 1)^2 \]

We will solve this equation for \( h \).

We start by using the Pythagorean theorem. Let \( h \) be the height of the tree. The distance from the point on the ground to the top of the tree is given as \( 2h + 1 \). The distance from the point on the ground to the base of the tree is 15 ft. Using the Pythagorean theorem, we set up the equation:

\[ 15^2 + h^2 = (2h + 1)^2 \]

Simplify the equation:

\[ 225 + h^2 = (2h + 1)^2 \]

Expanding the right-hand side:

\[ 225 + h^2 = 4h^2 + 4h + 1 \]

Rearrange the equation to bring all terms to one side:

\[ 225 + h^2 - 4h^2 - 4h - 1 = 0 \]

Simplify:

\[ -3h^2 - 4h + 224 = 0 \]

Multiply through by -1 to make the leading coefficient positive:

\[ 3h^2 + 4h - 224 = 0 \]

Solve the quadratic equation:

\[ h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where \( a = 3 \), \( b = 4 \), and \( c = -224 \).

Solving the quadratic equation, we get two solutions:

\[ h = \frac{-4 \pm \sqrt{16 + 2688}}{6} \]

\[ h = \frac{-4 \pm \sqrt{2704}}{6} \]

\[ h = \frac{-4 \pm 52}{6} \]

This gives us two potential solutions:

\[ h = \frac{48}{6} = 8 \quad \text{and} \quad h = \frac{-56}{6} \approx -9.3333 \]

Since the height of the tree cannot be negative, we discard \( h \approx -9.3333 \). Therefore, the valid solution is:

\[ h = 8 \]

Final Answer