A 10 foot ladder is leaning against a house. The bottom of the ladder is 5 feet from the building. How high is the top of the ladder? How do I round to the nearest hundredth?

We can regard the ladder, the floor and the house wall as forming a right angled triangle with the ladder representing the hypotenuse.
Using Pythagoras' Theorem and letting H be the height of the ladder then,
10² = 5² + H² H² = 100 - 25 = 75
H = √75 = 9.21954 (5dp)
As the answer is required to the nearest hundredth, then the rule for rounding off numbers is :-
"If the first figure to be discarded is 5 or more then the preceding figure is increased by 1".

As the first figure to be discarded is "9" (as this measures thousandths) which is greater than 5, then the preceding figure of 1 is increased by 1 to become 2.
Then H = 9.22 (2dp)