A painter sets a ladder up to reach the bottom of a second-story window 16 feet above the ground. The base of the ladder is 12 feet from the house. While the painter mixes the paint, a neighbor's dog bumps the ladder, which moves the base 2 feet farther away from the house. How far up the side of the house does the ladder reach/

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Wrong

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To find out how far up the side of the house the ladder reaches, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In this case, the ladder is the hypotenuse, and the distance from the base of the ladder to the house and the height of the window form the other two sides of the right triangle.

Let's call the distance from the base of the ladder to the house, which is initially 12 feet, as 'd1'. We can calculate the initial distance from the top of the ladder to the window using the Pythagorean theorem:

d1^2 + height^2 = ladder length^2

Substituting the values we know:

12^2 + height^2 = ladder length^2

144 + height^2 = ladder length^2

Now, let's calculate the final distance from the base of the ladder to the house, which is 2 feet longer than the initial distance. We'll call this new distance 'd2':

d2 = d1 + 2
d2 = 12 + 2
d2 = 14

Similarly, we can calculate the final ladder length using the Pythagorean theorem:

d2^2 + height^2 = ladder length^2

Substituting the values we know:

14^2 + height^2 = ladder length^2

196 + height^2 = ladder length^2

Now, let's compare the two Pythagorean equations we got for the initial and final ladder lengths:

144 + height^2 = 196 + height^2

The square of the heights cancels out on both sides of the equation:

144 = 196

This equation is not true, which means there is no possible height that makes the equation valid. In other words, moving the base of the ladder 2 feet farther away from the house made it impossible for the ladder to reach the window. Thus, the ladder does not reach up the side of the house.

Ladder is 12 feet out and 16 ft up,

A VERY UNSAFE WAY TO PUT THE LADDER!!

length of ladder --- L
L^2 = 12^2+16^2
= 400
L= √400 = 20

new position:
base = 14 , height = ? , L = 20

solve:
h^2 + 14^2 = 20^2

Base=14' Height=2 Length =16'