Question

A civil engineer is mapping the overhead clearance of his family’s property on a coordinate grid. The ground is represented by the x-axis and the base of the house is at the origin. There are two trees on the property. One tree is 10 feet from the base of the house and is 14 feet tall. The other tree is 13 feet from the base of the house and is 9 feet tall. What is the distance from the base of the house to the closest treetop? Round your answer to the nearest tenth.

Answer 1

15.8 feet

Explanation:

The base of the house is at the origin. This makes the ordered pair representing it (0, 0).

The top of the shorter tree is represented by the ordered pair (13, 9). The top of the taller tree is represented by the ordered pair (10, 14).

We will use the distance formula for each.

For the shorter tree:

`d=√((y_2-y_1)^2+(x_2-x_1)^2)\\\\=√((9-0)^2+(13-0)^2)\\\\=√(9^2+13^2)\\\\=√(81+169)\\\\=√(250)\approx 15.8`

For the taller tree:

`d=√((y_2-y_1)^2+(x_2-x_1)^2)\\\\=√((14-0)^2+(10-0)^2)\\\\=√(14^2+10^2)\\\\=√(196+100)\\\\=√(296)\approx 17.2`

The smaller distance is 15.8.

Answer 2

We are given
x1 = 10 ft
y1 = 14 ft
x2 = 13 ft
y2 = 9 ft

We are asked to find the distance between the base of the house to the closest treetop.

So,
d1 = √(10² + 14²) = 17.20
d2 √(13² + 9²) = 15.81

The distance is 15.81 ft to the closest treetop which is the tree 13 feet from the base of the house.