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. 1320 is not the hypotenuse.
I did this for you before guessing what you might have meant.
Tell me what b , c, and Ø are before I attempt it again.
To find the lengths of b and c, and the measure of angle theta, we can use the concept of right triangles and trigonometric ratios. Let's break down the problem step by step:
Step 1: Draw a diagram
Draw a diagram to represent the situation described in the problem. Label the vertical length as "b", the horizontal length as "c", and the angle between the cable and the horizontal as "theta".
/|
/ |
/ |
b / |
/ | c
/ |
/______|
Step 2: Identify the known information
From the problem statement, we know that:
- The cable rises 2 feet for every 5 feet of horizontal length.
- The top of the cable is fastened 1320 feet above the cable’s lowest point.
Step 3: Set up equations
Using the information given, we can set up the following equations:
Equation 1: tan(theta) = b / c
Equation 2: c^2 + b^2 = 1320^2 (from the Pythagorean theorem)
Step 4: Solve for b and c
Since we have two equations (Equation 1 and Equation 2) with two unknowns (b and c), we can solve them simultaneously.
Here's how you can solve it manually:
From Equation 1, we have:
tan(theta) = b / c
Rearranging, we get:
b = c * tan(theta)
Substitute this value of "b" into Equation 2:
c^2 + (c * tan(theta))^2 = 1320^2
Simplify the equation:
c^2 + c^2 * tan^2(theta) = 1320^2
Factor out c^2:
c^2 * (1 + tan^2(theta)) = 1320^2
Simplify further:
c^2 * (1 + tan^2(theta)) = 1320^2
c^2 * sec^2(theta) = 1320^2
Take the square root of both sides:
c * sec(theta) = 1320
c = 1320 / sec(theta)
Now, you can plug this value of "c" back into Equation 1 to find "b":
b = c * tan(theta)
Once you have the values of "b" and "c", you can find the value of the angle theta by using the inverse trigonometric function:
theta = arctan(b / c)
To find the actual values, you can use a scientific calculator or trigonometric tables.
Alternatively, you can use a numerical solver or a trigonometric calculator to perform the calculations for you.