The cheerleaders at a football game launch T-shirts into the crowd from the back of a golf cart that is 2 feet off the ground. The T-shirts have an upward velocity of 30 feet per second. Using the function , identify the y-intercept and explain its meaning.

Question

Use the explicit formula to identify:

What type of sequence is it? Arithmetic or Geometric?

Answer 1

To analyze the situation involving the cheerleaders launching T-shirts, we can create a model for the height \( h(t) \) of the T-shirts above the ground at any given time \( t \) (in seconds). The motion of the T-shirts can typically be modeled by a quadratic function due to the influence of gravity.

1. Determining the Height Function: Assuming the T-shirt is launched from a height of 2 feet (the height of the golf cart), and having an initial upward velocity of 30 feet per second, we can model the height \( h(t) \) of the T-shirt over time using the formula: \[ h(t) = -16t^2 + 30t + 2 \] Here, \( -16t^2 \) accounts for the downward effect of gravity (in feet per second squared), \( 30t \) represents the initial upward velocity, and \( 2 \) is the initial height from which the T-shirt is launched.

2. Identifying the y-intercept: The y-intercept of a function occurs when \( t = 0 \). To find the y-intercept for our function \( h(t) \): \[ h(0) = -16(0)^2 + 30(0) + 2 = 2 \] Thus, the y-intercept is \( 2 \).

Meaning of the y-intercept: The y-intercept of \( 2 \) feet means that at time \( t = 0 \) seconds (the moment the T-shirt is launched), its height is 2 feet above the ground, which is consistent with the height of the golf cart from which the T-shirt is launched.

3. Identifying the Type of Sequence: To determine whether the sequence represented in the height function describes an arithmetic or geometric sequence, consider the nature of the function:

• An arithmetic sequence involves a constant difference between sequential terms.
• A geometric sequence involves a constant ratio between sequential terms.

The height function \( h(t) = -16t^2 + 30t + 2 \) represents the height at various points in time, and since it is a quadratic function, its values increase to a maximum and then decrease as time continues due to the shaping nature of the parabolic trajectory.

The height values do not represent consistent differences or ratios. Thus, they do not fit into either definition for arithmetic or geometric sequences. This scenario describes a quadratic relation rather than a linear arithmetic or exponential geometric sequence.

In summary:

• Y-intercept: \( 2 \) (meaning initial height above ground at launch)
• Type of sequence: Not arithmetic or geometric; it's quadratic.

Answer 2

super small w/ work shown

Answer 3

Certainly! Let’s simplify the explanation and include the necessary work shown clearly.

Height Function

The height \( h(t) \) of the T-shirt at time \( t \) can be modeled by the function: \[ h(t) = -16t^2 + 30t + 2 \]

Y-Intercept

To find the y-intercept, we evaluate the function at \( t = 0 \): \[ h(0) = -16(0)^2 + 30(0) + 2 = 2 \] Y-Intercept: \( 2 \) feet

Meaning:

This means that at the moment of launch (when \( t = 0 \)), the T-shirt is \( 2 \) feet above the ground (the height of the golf cart).

Type of Sequence

The height function is a quadratic function due to the \( t^2 \) term:

• Arithmetic sequence: Constant difference between terms.
• Geometric sequence: Constant ratio between terms.

Since this function is quadratic (not linear and does not have constant differences or ratios), it does not represent either arithmetic or geometric sequences.

Summary:

• Y-Intercept: \( 2 \) feet (means initial launch height).
• Type of Sequence: Not arithmetic or geometric; it’s quadratic.

Answer 4

What is the common difference or common ratio?

Answer 5

In the context of the function we used to model the height of the T-shirts, since it is a quadratic function, it does not exhibit a linear common difference or a constant ratio that characterizes arithmetic or geometric sequences. However, if we still want to explore the nature of the changes in height over specific time intervals, we can calculate the "differences" in heights at successive time points, and the "ratios" at those heights.

Common Difference

To find a common difference, we could look at the heights at discrete time intervals. Let's calculate the height of the T-shirt at three time increments: \( t = 0 \), \( t = 1 \), and \( t = 2 \).

1. Calculate Heights:

   • \( h(0) = -16(0)^2 + 30(0) + 2 = 2 \)
   • \( h(1) = -16(1)^2 + 30(1) + 2 = -16 + 30 + 2 = 16 \)
   • \( h(2) = -16(2)^2 + 30(2) + 2 = -64 + 60 + 2 = -2 \) (note: the T-shirt hits the ground before reaching this height)

2. Calculate Common Differences:

   • From \( h(0) \) to \( h(1) \): \[ h(1) - h(0) = 16 - 2 = 14 \]
   • From \( h(1) \) to \( h(2) \): \[ h(2) - h(1) = -2 - 16 = -18 \]

Results

• Common Differences:
  • Between \( t = 0 \) and \( t = 1 \): \( 14 \)
  • Between \( t = 1 \) and \( t = 2 \): \( -18 \)

Since these differences are not constant, there is no common difference in the context of an arithmetic sequence.

Common Ratio

Next, let's assess any "common ratios." For this, we calculate: \[ \text{Ratio} = \frac{h(t+1)}{h(t)} \]

1. Calculate Ratios:
   • From \( h(0) \) to \( h(1) \): \[ \frac{h(1)}{h(0)} = \frac{16}{2} = 8 \]
   • From \( h(1) \) to \( h(2) \): \[ \frac{h(2)}{h(1)} = \frac{-2}{16} = -0.125 \]

Results

• Common Ratios:
  • Between \( t = 0 \) and \( t = 1 \): \( 8 \)
  • Between \( t = 1 \) and \( t = 2 \): \( -0.125 \)

Since these ratios are not constant, there is no common ratio in the context of a geometric sequence.

Conclusion

• Common Difference: Not consistent (14, then -18).
• Common Ratio: Not consistent (8, then -0.125).

In summary, due to the quadratic nature of the height function, there is no common difference or common ratio in the contexts of arithmetic and geometric sequences.

Answer 6

a short answer

Answer 7

Common Difference: Not consistent (14, then -18).

Common Ratio: Not consistent (8, then -0.125).

There are no constant common differences or ratios due to the quadratic nature of the height function.

Answer 8

What is the first term in the sequence?

Answer 9

The first term in the sequence, corresponding to the height of the T-shirt at the moment of launch (\( t = 0 \)), is:

First Term: \( h(0) = 2 \) feet.

This represents the initial height of the T-shirt above the ground when it is launched from the golf cart.

Answer 10

What is ?

Answer 11

It seems like your question got cut off. If you meant to ask about a specific variable or concept (for example, a term from the height function or something else related), please provide more context so I can assist you accurately!