A ladder leaning against a building makes a 40 degree angle with the ground. The top of the ladder reaches 10 feet above the ground. How tall is the ladder to the nearest whole foot?
10/length = sin 40
length = 10/sin40 = 10/0.642 = 15.56 ft
To find the height of the ladder, we can use the trigonometric ratio sine.
The sine of an angle is equal to the opposite side divided by the hypotenuse. In this case, the opposite side is the height of the building and the hypotenuse is the length of the ladder.
We know that the angle is 40 degrees and the height of the building is 10 feet, so we can set up the following equation:
sin(40°) = 10 / length of the ladder
To solve for the length of the ladder, we can rearrange the equation:
length of the ladder = 10 / sin(40°)
Using a calculator, we can find the approximate value of sin(40°) to be 0.643.
Therefore, the length of the ladder is:
length of the ladder = 10 / 0.643
length of the ladder ≈ 15.53 feet
So, the length of the ladder to the nearest whole foot is 16 feet.
To find the height of the ladder, we can use trigonometry.
First, let's label the sides of the triangle formed by the ladder, the building, and the ground. The ladder is the hypotenuse of the triangle, the height of the ladder is the side opposite the angle of 40 degrees, and the distance from the base of the ladder to the building is the adjacent side.
Using the trigonometric function tangent, we can set up an equation:
tan(40 degrees) = height of ladder / distance to the building
The tangent of an angle is equal to the length ratio of the side opposite the angle to the side adjacent to the angle.
Now we can rearrange the equation to solve for the height of the ladder:
height of ladder = tan(40 degrees) * distance to the building
Substituting the known values into the equation:
height of ladder = tan(40 degrees) * distance to the building
height of ladder = tan(40 degrees) * 10 feet
To calculate this, we can use a scientific calculator.
Using a calculator:
tan(40 degrees) = 0.8391
So,
height of ladder = 0.8391 * 10 feet
height of ladder ≈ 8.4 feet
Therefore, the height of the ladder to the nearest whole foot is 8 feet.