The cable supporting a ski lift rises 2 feet for each 5 feet of horizontal length. The top of the cable is fastened 1320 feet above the cable’s lowest point. Find the lengths b and c, and find the measure of the angle theta.

To solve this problem, we can use the concept of right triangles and trigonometric functions. Let's break down the problem step by step.

Step 1: Understanding the given information
- The cable supporting a ski lift rises 2 feet for each 5 feet of horizontal length.
- The top of the cable is fastened 1320 feet above the cable’s lowest point.

Step 2: Identifying the variables
- Let 'b' represent the horizontal length of the cable.
- Let 'c' represent the vertical rise of the cable.
- Let 'θ' represent the angle between the cable and the horizontal ground.

Step 3: Finding the length 'b'
Since the cable rises 2 feet for each 5 feet of horizontal length, we can set up a proportion based on the given information:
Vertical rise / Horizontal length = 2/5

Plugging in the given values:
c / b = 2/5

Simplifying the proportion:
2b = 5c

Step 4: Finding the length 'c'
We are told that the top of the cable is fastened 1320 feet above the cable’s lowest point. Therefore, the vertical rise 'c' is equal to 1320 feet.

c = 1320 feet

Step 5: Substituting the known values to find 'b'
We can substitute the value of 'c' into the equation we found earlier:

2b = 5c
2b = 5 * 1320
2b = 6600
b = 6600 / 2
b = 3300 feet

So the length 'b' is 3300 feet.

Step 6: Finding the angle 'θ'
The tangent of an angle is defined as the ratio of the opposite side to the adjacent side. In this case, the opposite side is the vertical rise 'c' and the adjacent side is the horizontal length 'b'.

Using the tangent function:
tan(θ) = c / b

Substituting the known values:
tan(θ) = 1320 / 3300

Now we can find the angle 'θ' by taking the inverse tangent (arctan) of both sides:
θ = arctan(1320 / 3300)

Using a calculator, we find that θ is approximately 21.57 degrees.

So the length 'b' is 3300 feet, the length 'c' is 1320 feet, and the angle 'θ' is approximately 21.57 degrees.

To find the lengths b and c, as well as the measure of the angle theta, we can use the given information about the rise and horizontal length.

Let's assign some variables to the given values:
Let x be the horizontal length.
Let b be the vertical length from the lowest point to point B.
Let c be the vertical length from point B to the top of the cable.

Based on the information given, we know that the cable rises 2 feet for every 5 feet of horizontal length. This gives us the following relationship:
b = (2/5) * x

We are also told that the top of the cable is fastened 1320 feet above the cable's lowest point, which means the total vertical length from the lowest point to the top is b + c. Therefore, we have:
b + c = 1320

Now, we can substitute the expression for b into the second equation:
(2/5) * x + c = 1320

To solve for x, we need to isolate it on one side of the equation. Let's rearrange the equation:
(2/5) * x = 1320 - c

To simplify this further, we can multiply both sides of the equation by 5/2:
x = (1320 - c) * (5/2)

Now, we have an equation for x in terms of c. Substituting this back into the first equation, we get:
b = (2/5) * [(1320 - c) * (5/2)]

Simplifying this expression further, we have:
b = (1320 - c) / 2

Therefore, we have expressions for both b and c.

To find the measure of angle theta, we can use basic trigonometry. The tangent of theta is equal to the ratio of the vertical distance (b + c) to the horizontal distance (x). So we have:
tan(theta) = (b + c) / x

Substituting the expressions for b and x:
tan(theta) = [(1320 - c) / 2 + c] / [(1320 - c) * (5/2)]

At this point, we have an equation for tangent(theta) in terms of c. We can then solve this equation to find the value of c. Once we find c, we can substitute it back into the expressions for b and theta to get their values.

To summarize:
- To find b, use b = (1320 - c) / 2.
- To find c, solve the equation for tangent(theta) = [(1320 - c) / 2 + c] / [(1320 - c) * (5/2)].
- To find theta, use the value of c in the expression for tangent(theta).

I hope this explanation helps you solve the problem!

let the rise be 2x , and the run be 5x, then the hypotenuse is 1320

(2x)^2 + (5x)^2 = 1320
29x^2 = 1320
x^2 = 1320/29
x = √(1320./29) = appr 6.74665

rise = 2x = appr 13.5 ft
run = 5x = appr 33.7 ft

don't know what your b and c refers to , but I am sure you can match them up with the answers given.
I will assume base angle is Ø

tan Ø = rise/run = 2/5
Ø = appr 21.8°