Problem: A cable from the top of a 60 foot high tower is to be attached to the ground (x) feet from the base of the tower.
A) if the cable makes an angle of t radians with the ground when attached, express t as a function of x.
B) if t = pi/5, how far is the end of the cable from the base of the tower?
60/x=tant
x=60/tant
To solve this problem, we can use trigonometry. Let's break down each part and figure out the equation for each question:
A) To express the angle t as a function of x, we can use the concept of tangent. Tangent is defined as the ratio of the opposite side to the adjacent side in a right triangle.
In this case, the tower makes a right triangle with the cable. The height of the tower is the opposite side, and the distance from the base of the tower to where the cable is attached (x) is the adjacent side.
So, we have:
tan(t) = opposite / adjacent
tan(t) = 60 / x
To solve for t, we need to take the arctan (inverse tangent) of both sides:
t = arctan(60 / x)
B) Now, if we are given t = pi/5, we can substitute this value into the equation we derived in part A.
t = pi/5
pi/5 = arctan(60 / x)
To solve for x, we can take the tangent of both sides:
tan(pi/5) = 60 / x
Multiplying both sides by x:
x * tan(pi/5) = 60
To find the value of x, we can use a calculator to evaluate the tangent of pi/5, then divide 60 by that value.
Hope this helps! Let me know if you have any more questions.