a 30 foot ladder leans against a wall at a 71 degree angle of elevztion find the distance of a ladder's base to the wall
Try to draw an image of the ladder leaning against the wall. The diagram will look like a right abgled triangle.
Now let the distance of the ladders base to the wall = x
Cos 71 = x / 30
x = 30Cos71
x = 9.8
To find the distance of the ladder's base to the wall, we can use trigonometry.
We know that the ladder forms a right angle with the wall, so we can use the sine function to relate the angle of elevation and the opposite side (the distance from the base to the wall).
The sine function is defined as sin(theta) = opposite / hypotenuse.
In this case, the angle of elevation is 71 degrees and the hypotenuse is the length of the ladder, which is 30 feet.
Let's calculate the distance from the base to the wall:
sin(71) = distance from base to wall / 30
To find the distance from the base to the wall, we can rearrange the equation:
distance from base to wall = sin(71) * 30
Using a calculator, we can find:
distance from base to wall ≈ 28.79 feet
So, the distance from the ladder's base to the wall is approximately 28.79 feet.
To find the distance of the ladder's base to the wall, we can use trigonometry. In this case, we have the length of the ladder (30 feet) and the angle of elevation (71 degrees).
The distance we need to find is the adjacent side in the right triangle formed by the ladder, the wall, and the ground.
We can use the cosine function, which is defined as the adjacent side divided by the hypotenuse. In this case, the adjacent side is the distance we need to find, and the hypotenuse is the length of the ladder.
So, the formula we'll use is:
cos(angle) = adjacent / hypotenuse
Let's substitute the known values into the formula:
cos(71 degrees) = adjacent / 30 feet
To isolate the adjacent side, we can rearrange the equation:
adjacent = cos(71 degrees) * 30 feet
Now we can use a calculator to find the cosine of 71 degrees:
cos(71 degrees) ≈ 0.314
Substituting this value back into the equation:
adjacent ≈ 0.314 * 30 feet
adjacent ≈ 9.42 feet
Therefore, the distance of the ladder's base to the wall is approximately 9.42 feet.