The problem states that we have a right triangle with legs of length $a$ and $b$, and a hypotenuse of length $c$. Given that $b = 2.5$ kilometers and $c = 4.6$ kilometers, we need to find the length of side $a$, rounded to the nearest tenth.

1. Problem Description

The problem states that we have a right triangle with legs of length $a$ and $b$, and a hypotenuse of length $c$. Given that $b = 2.5$ kilometers and $c = 4.6$ kilometers, we need to find the length of side $a$, rounded to the nearest tenth.

2. Solution Steps

We will use the Pythagorean theorem to solve this problem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In our case, the equation is:

$a^2 + b^2 = c^2$

We are given $b = 2.5$ and $c = 4.6$, so we can substitute these values into the equation:

$a^2 + (2.5)^2 = (4.6)^2$

Now, we need to solve for $a$:

$a^2 + 6.25 = 21.16$

$a^2 = 21.16 - 6.25$

$a^2 = 14.91$

$a = \sqrt{14.91}$

$a \approx 3.861347$

Since we need to round to the nearest tenth, we look at the hundredths place. Since the hundredths digit is 6 (which is greater than or equal to 5), we round up the tenths digit.

$a \approx 3.9$

3. Final Answer

a = 3.9 kilometers