Firefighting

(1) 2600

Explanation

We will use the Pythagorean theorem to find the distance between points \( A \) and \( B \).  If the distance is less than 3,000 yards, the crews can communicate by radio.  If it is greater than 3,000 yards, they cannot.

The line segments connecting points \( A \), \( B \), and \( C \) form a right triangle.  To find the distance \( c \) from \( A \) point to point \( B \), we can use the Pythagorean equation, substituting 2,400 for \( a \) and 1,000 for \( b \) and solving for \( c \).

This is the Pythagorean equation

\[ a^2 + b^2 = c^2 \]Substitute for \( a \) and \( b \)

\[ \require{color} \color{red}{2,400^2} + \color{blue}{1,000^2}=c^2 \]Evaluate each exponential expression

\[ 5,760,000 + 1,000,000 = c^2 \]Do the addition

\[ 6,760,000=c^2 \]Reverse the sides of the equation so that \( c^2 \) is on the left

\[ c^2=6,760,000 \]If \( c^2 \) = 6,760,000, then \( c \) must be a square root of 6,760,000.  Because \( c \) represents a length, it must be the positive square root of 6,760,000.

\[ c= \sqrt{6,760,000} \]Use a calculator to find the square root.

\[ c=2,600 \]The two crews are 2,600 yards apart.  Because this distance is less than the 3,000 yard range of the radios, then can communicate by radio.