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 side (let's call this height \( h \)), the ground between the foot of the ladder and the house is the horizontal side (length \( 15 \) feet), and the ladder itself is the hypotenuse (length \( 20 \) feet). So we draw a right triangle where one leg is vertical (representing the house), one leg is horizontal (15 feet, distance from ladder's foot to house), and the hypotenuse is the ladder (20 feet) 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 \( c \), \( a^{2}+b^{2}=c^{2} \). Let \( a = h \) (height on the house), \( b = 15 \) (distance from ladder to house), and \( c = 20 \) (length of ladder). So we have \( h^{2}+15^{2}=20^{2} \).

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)

Final Answers:

a) Diagram: A right triangle with horizontal leg 15 ft, hypotenuse 20 ft, vertical leg (height on house) \( h \).
b) \(\boxed{13.2}\)

Answer:

Part a) Diagram Explanation:

To model the situation, we can represent it as a right triangle. The house forms the vertical side (let's call this height \( h \)), the ground between the foot of the ladder and the house is the horizontal side (length \( 15 \) feet), and the ladder itself is the hypotenuse (length \( 20 \) feet). So we draw a right triangle where one leg is vertical (representing the house), one leg is horizontal (15 feet, distance from ladder's foot to house), and the hypotenuse is the ladder (20 feet) 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 \( c \), \( a^{2}+b^{2}=c^{2} \). Let \( a = h \) (height on the house), \( b = 15 \) (distance from ladder to house), and \( c = 20 \) (length of ladder). So we have \( h^{2}+15^{2}=20^{2} \).

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)

Final Answers:

a) Diagram: A right triangle with horizontal leg 15 ft, hypotenuse 20 ft, vertical leg (height on house) \( h \).
b) \(\boxed{13.2}\)