Question

At a point on the ground 35 ft from the base of a tree, the distance to the top of the tree is 1 ft more than 3 times the height of the tree. Find the height of the tree

Answer 1

12 ft

Explanation:

Let the height of the tree is h.

So the distance of top of the tree = 3 h + 1

Distance of base of tree = 35 ft

So, by use of Pythagoras theorem

`\left ( 3h+1 \right )^(2)=h^(2)+35^(2)`

`8h^(2)+6h-1224=0`

`4h^(2)+3h-612=0`

`h=(-3\pm √(9+4* 4* 612))/(8)`

`h=(-3\pm 99)/(8)`

Take positive sign

h = 12 ft

Thus, the height of tree is 12 ft.

Answer 2

Explanation:

This is a question that uses the Pythagorean Theorem.

a = 35 feet

b = x which is the height of the tree.

c = 3*x + 1 so we are trying to find x. Substitute into a b and c

a^2 + b^2 = c^2

35^2 + x^2 = (3x + 1)^2

35^2 + x^2 = 9x^2 + 6x + 1 Subtract x^2 from both sides.

35^2 = 8x^2 + 6x + 1 Subtract 35^2 from both sides.

0 = 8x^2 + 6x + 1 - 35^2

0 = 8x^2 + 6x - 1224

Does this factor?

(x + 12.75)(x - 12)

x - 12 = 0 is the only value that works.

x = 12

The tree is 12 feet high.

Note: I used the quadratic formula to solve this.