Brandon is making a ramp. The ramp stands 3 feet tall and is 9 feet long. How long is the base of the ramp?

Steps to Solve

1. Understand the Pythagorean theorem

We will use the Pythagorean theorem, which states that in a right triangle: $$ a^2 + b^2 = c^2 $$ where $a$ and $b$ are the legs of the triangle and $c$ is the hypotenuse.

2. Identify the known values

In this problem:

• The height of the ramp ( a = 3 ) feet
• The length of the ramp (hypotenuse) ( c = 9 ) feet

3. Set up the equation

We need to find the length of the base ( b ). Plugging the known values into the Pythagorean theorem equation, we get: $$ 3^2 + b^2 = 9^2 $$

4. Calculate ( 3^2 ) and ( 9^2 )

Calculating these gives: $$ 9 + b^2 = 81 $$

5. Solve for ( b^2 )

Subtract ( 9 ) from both sides: $$ b^2 = 81 - 9 $$ $$ b^2 = 72 $$

6. Calculate ( b )

To find ( b ), take the square root of ( 72 ): $$ b = \sqrt{72} $$

To simplify: $$ b = \sqrt{36 \times 2} $$ $$ b = 6\sqrt{2} $$

7. Find the approximate value of ( b )

Using a calculator, ( \sqrt{2} \approx 1.414 ): $$ b \approx 6 \times 1.414 \approx 8.485 $$