Question

20.) a wheelchair ramp with a length of 122 inches has a horizontal distance of 120 inches. a.) draw a diagram to model this situation. b.) what is the ramp’s vertical distance?

Explanation:

Part a)

To draw the diagram:

1. Draw a right triangle. The hypotenuse represents the length of the ramp (122 inches).
2. The horizontal leg represents the horizontal distance (120 inches).
3. The vertical leg represents the vertical distance (which we will find in part b). Label the hypotenuse as 122 inches, the horizontal leg as 120 inches, and the vertical leg as \( h \) (for height, the vertical distance we need to find).

Part b)

Step 1: Identify the formula

We can use the Pythagorean theorem for a right triangle, which is \( a^{2}+b^{2}=c^{2} \), where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two legs. Let the horizontal distance be \( a = 120 \) inches, the vertical distance be \( b=h \) (what we want to find), and the hypotenuse \( c = 122 \) inches.

Step 2: Substitute the known values into the formula

Substitute \( a = 120 \), \( c=122 \) into \( a^{2}+b^{2}=c^{2} \), we get \( 120^{2}+h^{2}=122^{2} \).

Step 3: Calculate the squares

Calculate \( 120^{2}=14400 \) and \( 122^{2} = 14884 \). So the equation becomes \( 14400+h^{2}=14884 \).

Step 4: Solve for \( h^{2} \)

Subtract 14400 from both sides of the equation: \( h^{2}=14884 - 14400=484 \).

Step 5: Solve for \( h \)

Take the square root of both sides. Since \( h \) represents a distance (length), we take the positive square root. \( h=\sqrt{484} = 22 \).

Answer:

a.) (Diagram: A right triangle with hypotenuse 122 in, horizontal leg 120 in, vertical leg \( h \))
b.) The ramp's vertical distance is 22 inches.