Question

21.) a 20-foot ladder is leaning against a house. the foot of the ladder is 15 feet from the house.
a.) draw a diagram to model this situation.

b.) how high will the ladder reach, to the nearest tenth of a foot?

Explanation:

Part (a) Diagram Explanation:

To model the situation, we can represent it as a right triangle. The house forms the vertical leg (let's call its length \( h \)), the ground from the house to the foot of the ladder is the horizontal leg (length \( 15 \) feet), and the ladder itself is the hypotenuse (length \( 20 \) feet). So, we draw a right triangle with one leg vertical (representing the house), one leg horizontal (representing the ground), and the hypotenuse (representing the ladder) connecting the top of the vertical leg to the end of the horizontal leg.

Part (b) Step-by-Step Solution:

Step 1: Identify the formula

We use the Pythagorean theorem, which states that for a right triangle with legs \( a \) and \( b \), and hypotenuse \( c \), \( a^{2}+b^{2}=c^{2} \). In this case, let \( a = h \) (height on the house), \( b = 15 \) (distance from house), and \( c = 20 \) (length of ladder). So the formula becomes \( h^{2}+15^{2}=20^{2} \).

Step 2: Solve for \( h^{2} \)

First, calculate \( 15^{2}=225 \) and \( 20^{2} = 400 \). Then, rearrange the formula to solve for \( h^{2} \): \( h^{2}=20^{2}- 15^{2}=400 - 225=175 \).

Step 3: Solve for \( h \)

Take the square root of both sides: \( h=\sqrt{175}\approx13.2 \) (rounded to the nearest tenth).

Answer:

(Part b):
The ladder will reach approximately \( 13.2 \) feet high.