Colleen is helping decorate for homecoming week.  She places the ladder against the wall at a safe angle. The ladder is 12 feet long and the base of the ladder is 3 feet away from the wall.What angle, in degrees, does the ladder make with the ground? Round to the nearest tenth.

To solve this problem, we can use the concept of trigonometry. Specifically, we'll use the tangent function, which is the ratio of the opposite side to the adjacent side in a right triangle.

Here, the ladder acts as the hypotenuse of a right triangle, the distance from the wall to the base of the ladder is the adjacent side, and the height from the ground to where the ladder touches the wall is the opposite side.

The formula for tangent of an angle (θ) is:

tan(θ) = opposite side / adjacent side

We know the length of the ladder (hypotenuse) is 12 feet and the distance from the wall (adjacent side) is 3 feet. We want to find the angle the ladder makes with the ground.

Rearranging the formula to solve for the angle gives us:

θ = arctan(opposite side / adjacent side)

We don't know the opposite side, but we can use the Pythagorean theorem to find it:

hypotenuse² = opposite side² + adjacent side²

Rearranging to solve for the opposite side gives us:

opposite side = sqrt(hypotenuse² - adjacent side²)

Substituting the given values:

opposite side = sqrt(12² - 3²) = sqrt(144 - 9) = sqrt(135)

Now we can substitute this value back into the formula for the angle:

θ = arctan(sqrt(135) / 3)

Using a calculator to find the arctan of this ratio and convert from radians to degrees (since the problem asks for the answer in degrees), we get:

θ ≈ 74.5 degrees

So, the ladder makes an angle of approximately 74.5 degrees with the ground.