David wants to find the height of a building. He places his 44 feet ladder against the very top of the building. David then measures the base of the building to his ladder which is 10 feet. How tall is the building? Round your answer to the nearest tenth.
We can use the Pythagorean theorem to solve this problem.
According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In this case, the length of the hypotenuse is the length of the ladder, which is 44 feet, and the length of one of the other sides is the height of the building. We can call the height of the building "x", and the length of the other side, which is the base of the building, is 10 feet. So, we can write the following equation:
x^2 + 10^2 = 44^2
x^2 + 100 = 1936
x^2 = 1936 - 100
x^2 = 1836
x ≈ √1836
x ≈ 42.9 feet
Therefore, the building is approximately 42.9 feet tall.
To find the height of the building, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, the ladder is the hypotenuse of the right triangle, and its height can be calculated using the formula:
ladder^2 = base^2 + height^2
Given that the ladder is 44 feet and the base is 10 feet, we can substitute these values into the formula:
44^2 = 10^2 + height^2
Simplifying the equation:
1936 = 100 + height^2
Now, isolate the variable height^2:
height^2 = 1936 - 100
height^2 = 1836
To find the height, take the square root of both sides:
height = sqrt(1836)
Using a calculator, the square root of 1836 is approximately 42.85.
Therefore, the height of the building is approximately 42.85 feet.
To find the height of the building, we can use a trigonometric ratio called tangent. In this case, we can use the tangent function:
tan(angle) = opposite / adjacent
Since David placed his ladder against the very top of the building, the ladder forms a right triangle with the height of the building as the opposite side and the distance from the base of the building to the ladder as the adjacent side.
Using the given values, we have:
tan(angle) = opposite / adjacent
tan(angle) = height / 10
We know that the angle is the angle between the ladder and the ground. We can find this angle by using the inverse tangent function or arctan.
angle = arctan(tan(angle))
Now we can substitute the values into our equation:
height = tan(angle) * 10
To find the value of angle, we can use the inverse tangent function by taking the arctan of the ratio of the height of the building to the distance of the ladder from the base of the building:
angle = arctan(height / 10)
Now we can plug in the values to calculate the angle:
angle = arctan(44 / 10)
angle ≈ arctan(4.4)
Using a calculator to find the arctan of 4.4, we get:
angle ≈ 78.69 degrees
Now we can substitute the angle into our equation to find the height of the building:
height = tan(78.69) * 10
Using a calculator to find the tangent of 78.69, we get:
height ≈ tan(78.69) * 10 ≈ 44 * 10 ≈ 440 feet
Therefore, the height of the building is approximately 440 feet.