Question

at a point on the ground 70 ft from the base of a tree, the distance to the top of the tree is 2 ft more than 3 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.)

Explanation:

Step1: Set up the Pythagorean theorem

Let the height of the tree be $h$. The distance from the point on the ground to the base of the tree is 70 ft and the distance from the point on the ground to the top of the tree is $3h + 2$. According to the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, we have $70^{2}+h^{2}=(3h + 2)^{2}$.

Step2: Expand the equation

$4900+h^{2}=9h^{2}+12h + 4$.

Step3: Rearrange to form a quadratic equation

$8h^{2}+12h - 4896=0$. Divide through by 4 to simplify: $2h^{2}+3h - 1224=0$.

Step4: Use the quadratic formula

The quadratic formula for $ax^{2}+bx + c = 0$ is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Here, $a = 2$, $b = 3$, and $c=-1224$. First, calculate the discriminant $\Delta=b^{2}-4ac=(3)^{2}-4\times2\times(-1224)=9 + 9792=9801$. Then $h=\frac{-3\pm\sqrt{9801}}{4}=\frac{-3\pm99}{4}$.

Step5: Find the valid solution

We have two solutions for $h$: $h_1=\frac{-3 + 99}{4}=\frac{96}{4}=24$ and $h_2=\frac{-3 - 99}{4}=\frac{-102}{4}=-25.5$. Since the height of a tree cannot be negative, we discard the negative solution.

Answer:

24