A 20 foot ladder is leaning against a building. If the ratio of the base of the ladder's distance from the building to its top's height from the ground id 3:4, find how high the top of the ladder is from the ground.
To find the height of the top of the ladder from the ground, we can set up a proportion using the given ratio.
Let's denote the distance from the base of the ladder to the building as "x" and the height from the ground to the top of the ladder as "y".
According to the given information, the ratio of x to y is 3:4. This can be expressed as:
x/y = 3/4
Now, we can use this proportion to find the value of y.
Cross-multiply the proportion:
4x = 3y
Divide both sides of the equation by 3:
(4x)/3 = y
Now, we know that the height from the ground to the top of the ladder (y) is (4x)/3.
We also know that the ladder is 20 feet long. Using the Pythagorean theorem, we can create another equation to represent the relationship between x, y, and the length of the ladder:
x^2 + y^2 = 20^2
Since we already know the value of y in terms of x, we can substitute it into the equation:
x^2 + ((4x)/3)^2 = 400
Multiply ((4x)/3)^2:
x^2 + (16x^2)/9 = 400
Combine like terms:
(9x^2 + 16x^2)/9 = 400
Multiply both sides by 9 to eliminate the denominator:
9x^2 + 16x^2 = 3600
Combine like terms:
25x^2 = 3600
Divide both sides by 25:
x^2 = 3600/25
x^2 = 144
Take the square root of both sides to solve for x:
x = ±√144
Since the distance cannot be negative, we take the positive square root:
x = √144
x = 12
Now that we have solved for x, we can substitute it into the equation for y:
y = (4x)/3
y = (4(12))/3
y = 16
Therefore, the height of the top of the ladder from the ground is 16 feet.