1. What are the degrees in Fahrenheit if it is 31 degrees Celsius? F = 1.8c + 32
2. A ping pong ball is released from a height of 60 centimetres 
and bounces to a height that is 
the previous height. What function estimates the height, 
, in cm of the ping pong ball after 
bounces?
Enter a number in each empty box to correctly complete the function.
The correct answer and explanation is:
1. Converting Celsius to Fahrenheit:
The formula for converting Celsius to Fahrenheit is:
[
F = 1.8C + 32
]
Given that the temperature in Celsius is 31°C, we can substitute this value into the formula:
[
F = 1.8(31) + 32
]
[
F = 55.8 + 32
]
[
F = 87.8
]
So, 31°C is equal to 87.8°F.
2. Estimating the Height of the Ping Pong Ball After Bounces:
Given the scenario where a ping pong ball is released from a height of 60 cm and bounces to half the height of the previous bounce, we need to write a function that estimates the height of the ball after ( n ) bounces.
The pattern in this problem is that after each bounce, the ball reaches half of the previous height. Therefore, the height of the ball after the first bounce is 30 cm, after the second bounce it’s 15 cm, and so on. This suggests that the height of the ball follows a geometric sequence, where each term is half of the previous one.
The general formula for the height ( h(n) ) after ( n ) bounces can be expressed as:
[
h(n) = h_0 \times r^n
]
Where:
- ( h_0 ) is the initial height (60 cm),
- ( r ) is the common ratio (in this case, 0.5, since the ball bounces to half the previous height),
- ( n ) is the number of bounces.
So, the function that estimates the height of the ball after ( n ) bounces is:
[
h(n) = 60 \times (0.5)^n
]
Explanation:
- The Initial Height: The ball starts at 60 cm, which is why this is the value we multiply by in the formula.
- The Common Ratio (0.5): Since each bounce reaches half the previous height, the ratio between each bounce is 0.5. This is the defining characteristic of a geometric sequence where each term is a fixed multiple of the previous one.
- Exponentiation for Bounces: The number of bounces is represented by ( n ), which affects the exponent of the common ratio (0.5). As the number of bounces increases, the height decreases exponentially, meaning the ball bounces to a smaller height each time.
- Practical Application: This function models the real-world scenario of a ball losing height with each bounce due to energy loss. In physics, this kind of situation can be modeled by exponential decay.
By plugging different values for ( n ) into this formula, we can predict the height of the ball after any given number of bounces. For example, after 3 bounces, the height will be:
[
h(3) = 60 \times (0.5)^3 = 60 \times 0.125 = 7.5 \text{ cm}
]
This approach ensures an accurate prediction based on the given parameters of the problem.