A ladder 35.4 feet long is leaning against an embarkment inclined 62.5 degrees to the horizontal. If the bottom of the ladder is 10.2 feet from the embarkment, what is the distance from the top of the ladder down the embarkment to the ground?
If the ladder makes an angle A with the ground and B with the embankment, then using the law of sines,
35.4/sin 117.5 = 10.2/sinB = x/sinA
Now you can find B, and A = 180-(117.5+B) and you can find x.
or, having found B, you can use the law of cosines:
x^2 = 35.4^2 + 10.2^2 - 2(35.4)(10.2)cosB
To find the distance from the top of the ladder down the embankment to the ground, we can use trigonometric ratios.
In this case, the ladder forms a right triangle with the embankment and the ground. The length of the ladder is the hypotenuse of the triangle, the distance from the bottom of the ladder to the embankment is the adjacent side, and the distance from the top of the ladder to the ground is the opposite side.
Given:
Length of the ladder (hypotenuse) = 35.4 feet
Distance from the bottom of the ladder to the embankment (adjacent side) = 10.2 feet
Angle between the embankment and the ground = 62.5 degrees
To find the distance from the top of the ladder down the embankment to the ground, we can use the trigonometric ratio tangent:
tan(angle) = opposite/adjacent
Let's substitute the known values:
tan(62.5 degrees) = opposite/10.2 feet
Solving for opposite:
opposite = tan(62.5 degrees) x 10.2 feet
Using a calculator:
opposite ≈ 19.515 feet
Therefore, the distance from the top of the ladder down the embankment to the ground is approximately 19.515 feet.
To solve this problem, we can use trigonometry, specifically the sine function. Let's break down the information given:
- The ladder is 35.4 feet long.
- The bottom of the ladder is 10.2 feet away from the embarkment.
- The embarkment is inclined 62.5 degrees to the horizontal.
We need to find the distance from the top of the ladder down the embarkment to the ground.
First, let's draw a diagram to visualize the situation. Imagine a right-angled triangle formed by the ladder, the embarkment, and the ground:
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|
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h | \ <-- Distance from top of ladder to the ground
| \
|___\
b
Here, 'h' represents the height from the top of the ladder down to the ground, and 'b' represents the distance from the bottom of the ladder to the embarkment.
Using trigonometry, we can relate the angles and sides of this triangle. The sine function will be helpful in this case.
The sine of an angle is defined as the ratio of the opposite side to the hypotenuse. In our case, the angle is 62.5 degrees, and the opposite side is 'h', while the hypotenuse is the length of the ladder (35.4 feet).
Using the sine function:
sin(62.5) = h / 35.4
To find 'h', we rearrange the equation:
h = sin(62.5) * 35.4
Now, let's calculate it:
h = sin(62.5) * 35.4
h ≈ 31.2425 feet
Therefore, the distance from the top of the ladder down the embarkment to the ground is approximately 31.2425 feet.