a ladder reaches 18 feet up the side of a building with its base 5 feet from the building. Find the slope of the ladder.
Let θ=angle of ladder with horizontal
tan(θ)=18/5
Solve for θ.
To find the slope of the ladder, we can use the Pythagorean theorem, which 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 other two sides.
In this case, the ladder forms the hypotenuse of the right triangle, while the vertical distance it reaches up the building forms one of the other sides. The horizontal distance from the base of the ladder to the building forms the remaining side.
According to the Pythagorean theorem, the equation for this situation would be:
(hypotenuse)^2 = (vertical side)^2 + (horizontal side)^2
Let's substitute the given values into the equation:
(18 feet)^2 = (vertical side)^2 + (5 feet)^2
324 = (vertical side)^2 + 25
Now, rearrange the equation to isolate the term representing the vertical side:
(vertical side)^2 = 324 - 25
(vertical side)^2 = 299
Taking the square root of both sides, we find:
vertical side = √299
So, the vertical side of the triangle, which represents the height to which the ladder reaches on the building, is approximately equal to √299 feet.
The slope of the ladder can be found by dividing the height (vertical side) by the horizontal distance. In this case, the horizontal distance is given as 5 feet.
slope = (vertical side) / (horizontal side)
slope = (√299 feet) / (5 feet)
And that gives us the slope of the ladder.