A swing has a maximum height of 2 m and a minimum height of 0.5 m and it takes 3 seconds to get from the maximum height to the minimum height. Approximate the instantaneous rate of change 1 second into the fall using the difference quotient when h is 0.01.

a)-1.3650
b)-0.1973
c)-0.6823
d)-1.0856

What is the arc length if the central angle is 225o and the radius of a circle is 3 cm?

a)2.25pi cm
b)2.4pi cm
c)0.4166 cm
d)3.75pi cm

A person waves his hand back and forth, where their finger tips move past the starting position 4 cm in each direction. If he waves 20 times every 15 s, what sine equation models the movement of the hand if the hand starts in the upright position and the right direction will be taken as positive?

a)y=8sin((30pi/20)t)
b)y=4sin((30pi/20)t)
c)y=4sin((40pi/15)t)
d)y=4sin((20pi/15)t)

first one:

a = 1.25
period = 6
2π/k = 6
6k = 2π
k= π/3

so we have y = 1.25 cos ((π/3)(t + d) + 0.75
we have to know the height at the beginning (t=0)
it was not given.
That is, where is the swing at the "start" ?

2.
circumference = 2π(3) = 6π
arc/6π = 225/360
arc = 6π(5/8) = 3.75π

3. amplitude = 4
period:
20 waves in 15 seconds
so 1 wave = 15/20 = 3/4
3/4 = 2π/k
3k = 8π
k = 8π/3 or 40π/15
looks like c)
y = 4sin ( (40π/15)t )

To find the instantaneous rate of change at 1 second into the fall, we can use the difference quotient. The difference quotient formula is given by:

f'(x) = [f(x+h) - f(x)] / h

In this case, the function representing the height of the swing over time is not given, but we know the maximum height is 2 m and the minimum height is 0.5 m. We also know that it takes 3 seconds to go from the maximum height to the minimum height.

Using this information, we can find the average rate of change over the interval of 2 seconds to 1 second before the fall (which is 1 second into the fall). Let's call the average rate of change "m".

m = [f(2) - f(1)] / (2-1)

Now, let's substitute the given values into the equation:

m = [2 - 0.5] / (2-1)
m = 1.5 / 1
m = 1.5

Therefore, the average rate of change at 1 second into the fall is 1.5 m/s.

However, the options provided do not match this result, so it seems there might be an error in the given choices.

Regarding the second question:

The arc length of a circle is given by the formula:

s = θr

where s is the arc length, θ is the central angle in radians, and r is the radius of the circle.

In this case, the central angle is given as 225° and the radius is 3 cm.

First, we need to convert the central angle from degrees to radians:

θ = 225° * (pi/180°)
θ = 5π/4 radians

Now, we can calculate the arc length:

s = (5π/4) * 3
s = 15π/4 cm

Therefore, the arc length is 15π/4 cm, which is approximately 3.75π cm.

Hence, the correct answer is d) 3.75π cm.

For the third question:

The equation for describing the sine function is given by:

y = Asin(Bx + C) + D

where A, B, C, and D are constants that determine the properties of the sine function.

In this case, we are given that the finger tips move 4 cm in each direction, which means the amplitude A is 4. We are also given that the person waves 20 times every 15 seconds, which means the period T is 15/20 = 0.75 seconds. The period is determined by the value of B in the equation, where T = (2π/B).

Therefore, B = 2π/T = 2π/0.75 = 8π/3.

Finally, since the hand starts in the upright position and the right direction is positive, the phase shift C is 0 and the vertical shift D is 0.

Putting it all together, the sine function that models the hand movement is:

y = 4sin((8π/3)t)

Hence, the correct answer is c) y = 4sin((40π/15)t).

To find the approximate instantaneous rate of change 1 second into the fall using the difference quotient, we'll need to use the formula:

Instantaneous rate of change = (f(x + h) - f(x)) / h

In this case, the function f represents the height of the swing at time x, and h represents a small change in time. Given that it takes 3 seconds for the swing to go from the maximum height to the minimum height, we can assume that the swing is at its maximum height at x=0 and at its minimum height at x=3.

Let's calculate the values of f(1) and f(0) to find the approximate instantaneous rate of change at x=1. We'll use h = 0.01 as given.

f(1) = (maximum height - minimum height) * ((1 - 0)/(3 - 0)) + minimum height
= (2 - 0.5) * (1/3) + 0.5
= 1.5 * (1/3) + 0.5
= 0.5 + 0.5
= 1

f(0) = maximum height = 2

Now, let's calculate the approximate instantaneous rate of change:

Rate of change = (f(1 + 0.01) - f(1)) / 0.01
= (f(1.01) - 1) / 0.01

You can plug in the value of f(1.01) using the same formula:

f(1.01) ≈ (maximum height - minimum height) * ((1.01 - 0)/(3 - 0)) + minimum height

After calculating f(1.01), substitute the value into the rate of change formula:

Rate of change ≈ (f(1.01) - 1) / 0.01

Evaluate this expression and choose the correct option from the given choices.

To find the arc length of a circle given the radius and central angle, we can use the formula:

Arc length = (angle / 360) * (2 * π * radius)

In this case, the central angle is given as 225 degrees and the radius is given as 3 cm. Substituting these values into the formula, we have:

Arc length = (225 / 360) * (2 * π * 3)
= (5/8) * (2 * π * 3)
= (5/8) * (6π)
= 30π/8

Simplifying further, we get:

Arc length = 15π/4

Therefore, the correct option is:

a) 2.25π cm

To model the movement of the hand as a sine function, we can use the formula:

y = A * sin(B * t)

Where A represents the amplitude, B represents the frequency, and t represents time.

In this case, the hand moves 4 cm in each direction, so the amplitude A would be 4. The hand waves 20 times in 15 seconds, so the frequency can be calculated as:

Frequency = (Number of waves) / (Time taken)
= 20 / 15
= 4/3

Therefore, the frequency B would be (4π/3). Since the function starts in the upright position and the right direction is positive, the phase shift would be 0.

The equation that models the movement of the hand would be:

y = 4 * sin((4π/3) * t)

Therefore, the correct option is:

b) y = 4sin((30π/20)t)