A ladder 35 foot long is leaning against an embankment, inclined 62.5 degrees to the horizontal. if the bottom of the ladder is 10.2 feet from the embankment, what is the distance from the top of the ladder down to the embankment to the ground?

Draw a diagram. If we label the triangle ABC, with sides a,b,c then the law of sines tells us

sinC/10.2 = sin117.5°/35
Now you have angles B and C, so you know A=180-(B+C)

a/sinA = sin117.5°/35
or
a^2 = 35^2+10.2^2 - 2*35*10.2 cosA

A 38 ft long ladder is leaning against an embankment inclined at 60 deg. to the horizontal. If the bottom of the ladder is 15 ft from the embankment, what is the distance from the top of the ladder down to the embankment to the ground?

To find the distance from the top of the ladder down to the embankment to the ground, we can use trigonometry.

Let's call the distance from the top of the ladder down to the embankment "x".

We can see that the ladder, the embankment, and the ground form a right triangle. The angle between the ladder and the ground is the same as the angle between the ladder and the embankment, which is 62.5 degrees.

We can use the sine function to find the value of x:

sin(62.5 degrees) = x / 35 feet

To solve for x, we can rearrange the equation:

x = 35 feet * sin(62.5 degrees)

Using a calculator, we can find that sin(62.5 degrees) is approximately 0.9004.

x = 35 feet * 0.9004

x is approximately equal to 31.5134 feet.

Therefore, the distance from the top of the ladder down to the embankment to the ground is approximately 31.5134 feet.

To find the distance from the top of the ladder down to the embankment to the ground, we can use trigonometry.

Let's break down the problem into two right-angled triangles: one formed by the ladder, the embankment, and the ground, and the other formed by the ladder, the embankment, and the distance from the top of the ladder down to the embankment to the ground.

In the first triangle, the angle between the ladder and the ground is 90 degrees, and the angle between the ladder and the embankment is 62.5 degrees. The distance from the bottom of the ladder to the embankment is given as 10.2 feet.

Using trigonometry, we can determine the length of the embankment:

embankment = length of ladder * cosine of the angle between the ladder and the embankment

embankment = 35 * cos(62.5)

embankment ≈ 35 * 0.4719

embankment ≈ 16.5075 feet

Now, let's move to the second triangle. We want to find the distance from the top of the ladder down to the embankment to the ground. We already know the length of the embankment (16.5075 feet), and the angle between the ladder and the embankment is still 62.5 degrees.

Using trigonometry again, this time we can determine the distance from the top of the ladder down to the embankment to the ground:

distance = length of embankment * sine of the angle between the ladder and the embankment

distance = 16.5075 * sin(62.5)

distance ≈ 16.5075 * 0.8808

distance ≈ 14.53 feet

Therefore, the distance from the top of the ladder down to the embankment to the ground is approximately 14.53 feet.