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(1) You want to hang a 600 pound statue from your ceiling for a party. It will be hung by two cables each making a 60 degree angle with the ceiling. How much tension will be in each of the cables? Round your answer to the nearest pound.

(2) Find all solutions for the equation of 3cos(t)+4=2 on the interval [0, π), or answer "N/A" if there is no solution.

(3) Consider the numbers 0, 1, 2, 3, and 4. Take the square root of each of these numbers, then divide each by 2. Describe the relationship between the values you receive and the trigonometric ratios.

(4) A Ferris wheel boarding platform is 4 meters above the ground, has a diameter of 66 meters, and makes one full rotation every 5 minutes. How many minutes of the ride are spent higher than 47 meters above the ground?

1 Answer

4 votes

that is physics ...

but yes, applied math. we need to know the formulas though.

(1)

in general, since there are 2 cables supporting in an equal way.

that means each cable is responsible for 600/2 = 300 pounds to bring and hold up.

a cable or rope at an angle has to handle a combined tension force : horizontally (Fx) and vertically (Fy).

the tension force (Ftens) on the rope is a combination of both.

we know

Fx = Ftens × cos(theta)

Fy = Ftens × sin(theta)

from the problem we do know Fy (the vertical = up/down force), as this is the force needed to lift and keep the 300 pound weight up there.

and that is Fgravity, the force needed to counteract gravity.

Fgravity = mass × g

g being the constant gravitational acceleration of Earth = 9.8 m/s²

forces are described in Newton.

1 N ≈ 0.225 pounds (lifting on Earth)

so, to lift 1 pound requires 1/0.225 ≈ 4.44822 N

to lift 300 pounds requires

4.44822 × 300 ≈ 1334.47 N

that is what Fy is for one of the 2 cables.

the tension on one of the cables is then given by

Fy = Ftens × sin(60)

Ftens = Fy / sin(60) = 1334.47 / sin(60) =

= 1,540.913227... N = 346.41107515867... pounds

≈ 346 pounds per cable.

(2)

3cos(t) + 4 = 2

3cos(t) = -2

cos(t) = -2/3

cosine is negative in the 2nd and 3rd quadrant.

so, for t > pi/2 and t < 3pi/2.

because the given interval is [0, pi), we are only looking at the 2nd quadrant (pi/2, pi).

t = 131.8103149...° = 2.300523983... rad

(3)

well, that are the numbers

1/2

sqrt(2)/2 = 1/sqrt(2)

sqrt(3)/2

1

they are getting bigger and bigger, all positive, so they indicate larger and larger angles

1/2 is :

sin(30° or pi/6 or 150° or 5pi/6)

cos(60° or pi/3 or 300° or 5pi/3)

1/sqrt(2) is :

sin(45° or pi/4 or 135° or 5pi/4)

cos(45° or pi/4 or 315° or 7pi/4)

sqrt(3)/2 is :

sin(60° or pi/3 or 120° or 2pi/3)

cos(30° or pi/6 or 330° or 11pi/6)

1 is :

sin and csc(90° or pi/2)

cos and sec(0° or 0pi or 360° or 2pi)

tan and cot(45° or pi/4 or 225° or 5pi/4)

(4)

the height moves between 4 meters and 70 meters in a circle.

the circumference of the circle is 2pi×r or pi×d, so in our case : 66pi meters.

it takes 5 minutes to move along these 66pi meters.

let's say, when the height is 4 meters (starting position), the angle is 0 and the arc is 0.

after a quarter trip the angle is 90° or 66pi/4, and the height is 4 + 66/2 = 37 meters

and at 70 meters the angle is 180° or 66pi/2.

the function of the height based on the current angle is then for the first half-circle

height(theta) = 4 + (theta/360)×2×66

or

height(theta) = 4 + (theta/(2pi))×2×66

now we need to find the angle theta for which we reach the height of 47 meters :

47 = 4 + (theta/360)×132

43 = (theta/360)×132

theta/360 = 43/132

theta = 360×43/132 = 117.2727272...°

= 2.046795214... rad

so, after starting at the lowest position at 4 meters we reach the height of 47 meters at an angle of about 117°.

then we get and stay above 47 meters until we get to

360 - theta = 242.7272727...°

= 4.236390093... rad

when going down again on the second half-circle of the trip.

that means we are at and above 47 meters for

(360 - theta) - theta = 360 - 2×theta = 125.4545455...°

= 2.18959488... rad

of the whole trip of 360° or 2pi. which takes 5 minutes.

the time we spend there is then

5 × (360 - 2×theta)/360 = 1.742424242... minutes

= 1 minute 44.54545454... seconds

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