Questions: Use the Pythagorean Theorem and the square root property to solve the following problem. Express your answer in simplified radical form. Then find a decimal approximation to the nearest tenth. A rectangular park is 30 miles long and 10 miles wide. How long is a pedestrian route that runs diagonally across the park? In simplified radical form, the pedestrian route is miles long Rounded to the nearest tenth, the pedestrian route is miles long

Solution Steps

The Pythagorean Theorem states that for a right-angled triangle, \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse. Here, the length and width of the rectangular park form the two perpendicular sides of the triangle, and the diagonal is the hypotenuse.

Given:

• Length (\( a \)) = 30 miles
• Width (\( b \)) = 10 miles

Substitute the values into the theorem: \[ c^2 = 30^2 + 10^2 \]

Calculate the squares of the given lengths: \[ 30^2 = 900 \] \[ 10^2 = 100 \]

Add the results: \[ c^2 = 900 + 100 = 1000 \]

Take the square root of both sides to solve for \( c \): \[ c = \sqrt{1000} \]

Simplify the radical: \[ \sqrt{1000} = \sqrt{100 \times 10} = 10\sqrt{10} \]

Calculate the decimal value of \( 10\sqrt{10} \) to the nearest tenth: \[ \sqrt{10} \approx 3.162 \] \[ 10 \times 3.162 = 31.62 \]

Rounded to the nearest tenth: \[ 31.62 \approx 31.6 \]

Final Answer