An extension ladder adjusts in length from 10 ft to 16 ft. Suppose you've leaned your ladder against your house sot that, at its minimum lenght, it touches the base of your first floor window, 9 feet above the ground. Without moving the base, how high will the ladder reach when extended (to one decimal place)?
x^2+9^2 = 10^2 where x is base to house along ground
x^2 = 100 - 81 = 19
x = sqrt(19)
then
x^2 + h^2 = 16^2
h^2 = 256 - 19 = 237
h = sqrt (237)
To find out how high the ladder will reach when extended, we can use the concept of similar triangles.
First, let's set up the ratio of the corresponding sides of the similar triangles formed by the ladder, the ground, and the height of the ladder when extended:
Extended ladder length (16 ft) / Vertical distance covered by the ladder (x) = Initial ladder length (10 ft) / Vertical distance covered by the ladder (9 ft +x)
Simplifying the equation, we have:
16 / x = 10 / (9 + x)
To solve for x, we can cross multiply:
16 * (9 + x) = 10 * x
Expand the equation:
144 + 16x = 10x
Combine like terms and subtract 10x from both sides:
6x = 144
Divide both sides by 6:
x = 24
Therefore, the ladder will reach a height of 24 feet when extended.