A 10 foot ladder is to be placed against the side of a building. The base of the ladder must be placed at an angle of 72degree with the level ground for a secure footing. Find to the nearest inch, how far of the ladder should be from the side of the building and how far up the side of the building the ladder will reach
?
The distance from the side of the building to the base of the ladder should be 8.7 inches. The ladder will reach up the side of the building 9 feet.
To find out how far the ladder should be from the side of the building and how far up the side of the building it will reach, we can use trigonometric ratios.
Let's define:
- Length of the ladder = 10 feet
- Angle between the ladder and the ground = 72 degrees
First, we need to find the length of the ladder that is on the ground (base) and the length of the ladder that is along the side of the building (height).
Using the trigonometric ratio for sine, we can find the height:
sin(angle) = opposite/hypotenuse
sin(72) = height/10
height = sin(72) * 10
Next, we need to find the length of the ladder on the ground, which is the adjacent side. We can use the trigonometric ratio for cosine:
cos(angle) = adjacent/hypotenuse
cos(72) = base/10
base = cos(72) * 10
Now, let's calculate the values:
height = sin(72) * 10 = 9.3969 feet (to the nearest tenth)
base = cos(72) * 10 = 3.0777 feet (to the nearest tenth)
Therefore, the ladder should be placed approximately 3.1 feet from the side of the building, and it will reach up approximately 9.4 feet along the side of the building.
To solve this problem, we can use trigonometry, specifically the trigonometric functions sine, cosine, and tangent.
Let's label the distance that the ladder should be from the side of the building as 'x', and the distance that the ladder will reach up the side of the building as 'y'.
Since we have an angle and the length of the ladder, we can use the sine function, as sine(theta) = opposite/hypotenuse.
In this case, the opposite side is 'y', the hypotenuse is the length of the ladder (10 feet), and the angle is 72 degrees. We can write the equation as:
sin(72 degrees) = y/10
To find the value of 'y', we can rearrange the equation:
y = 10 * sin(72 degrees)
Using a scientific calculator, we can evaluate sin(72 degrees) to get the value of 'y'.
After finding 'y', we can use the cosine function, as cosine(theta) = adjacent/hypotenuse, to find 'x'. In this case, the adjacent side is 'x' and the hypotenuse is still 10 feet. We can write the equation as:
cos(72 degrees) = x/10
To find the value of 'x', we can rearrange the equation:
x = 10 * cos(72 degrees)
Using a scientific calculator, we can evaluate cos(72 degrees) to get the value of 'x'.
Finally, we round the values of 'x' and 'y' to the nearest inch to find how far the ladder should be from the side of the building and how far up the side of the building the ladder will reach.