1. A hill with a 35 degree grade (incline with horizontal) is cut down for a roadbed to a 10 degree grade. If the distance from the base to the top of the original hill is 800 ft., How many vertical ft will be removed from the hill rounded to the nearest ft

To find out how many vertical feet will be removed from the hill, we need to calculate the difference between the heights of the original hill and the new roadbed.

Let's break down the problem and solve it step by step:

Step 1: Calculate the height of the original hill.
We are given that the distance from the base to the top of the original hill is 800 ft, and the hill has a 35 degree grade. To find the height, we will use trigonometry. The formula to calculate the height (h) given an angle (θ) and a distance (d) is:

h = d * sin(θ)

Plugging in the values:
h = 800 ft * sin(35 degrees)
h ≈ 456.83 ft

So, the height of the original hill is approximately 456.83 ft.

Step 2: Calculate the height of the new roadbed.
Since the roadbed has a 10 degree grade, we can use the same formula as before to calculate the new height. The distance (d) will remain the same at 800 ft, and the angle (θ) is now 10 degrees.

h = 800 ft * sin(10 degrees)
h ≈ 139.16 ft

So, the height of the new roadbed is approximately 139.16 ft.

Step 3: Calculate the vertical feet removed.
To find the difference in height, we subtract the height of the new roadbed from the height of the original hill:

Vertical feet removed = height of original hill - height of new roadbed
Vertical feet removed ≈ 456.83 ft - 139.16 ft
Vertical feet removed ≈ 317.67 ft

Rounded to the nearest whole foot, approximately 318 vertical feet will be removed from the hill.

Therefore, the answer is 318 ft (rounded to the nearest foot).