91Ó°ÊÓ

Write a formala to express each of the following sentences: a. The sale price is \(20 \%\) off the original price. Use \(S\) for sale price and \(P\) for original price to express \(S\) as a function of \(P\). b. The time in Paris is 6 hours ahead of New York. Use \(P\) for Paris time and \(N\) for New York time to express \(P\) as a function of \(N\). (Represent your answer in terms of a 12 -hour clock.) How would you adjust your formula if \(\bar{P}\) comes out preater than \(12 ?\) c. For temperatures above \(0^{\circ} \mathrm{F}\) the wind chill effect can be estimated by subtracting two-thirds of the wind speed (in miles per hour) from the outdoor temperature. Use \(C\) for the effective wind chill temperature, \(W\) for wind speed, and \(T\) for the actual outdoor temperature to write an equation expressing \(C\) in terms of \(W\) and \(T\).

Step by step solution

01

Title - Express sale price as a function of original price

The sale price is given as 20% off the original price. This means the sale price is 80% of the original price. To find S as a function of P, multiply the original price by 0.80. Formula: \[ S = 0.80P \]

02

Title - Express Paris time as a function of New York time

Paris time is always 6 hours ahead of New York time. To express the Paris time (P) as a function of New York time (N), add 6 to the New York time. Formula: \[ P = N + 6 \]

03

Title - Adjust the formula for 12-hour clock

If P (Paris time) calculated in the previous step is greater than 12, subtract 12 to adjust it to a 12-hour clock. Use the modulo operation to represent time correctly in a 12-hour format. Formula: \[ P = (N + 6) \mod 12 \]

04

Title - Express effective wind chill temperature in terms of wind speed and actual temperature

The effective wind chill temperature is calculated by subtracting two-thirds of the wind speed from the actual outdoor temperature. To express C as a function of W and T, use the equation. Formula: \[ C = T - \frac{2}{3}W \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Notation

Function notation is a way to clearly define the relationship between an input and an output. It is often written using letters like f(x) where 'f' represents the function, and 'x' is the variable input.
In the context of these exercises, function notation helps us express how one quantity depends on another. For example, if S is the sale price and P is the original price, we can use function notation to write S as a function of P.
In our problem, the sale price S is 80% of the original price P. Using function notation, we write this relationship as follows:

$$ S = 0.80P $$
This compactly shows that S depends on P, making calculations and interpretations straightforward.

Percentage Calculations

Percentage calculations are essential in many real-world scenarios, especially dealing with discounts and growth rates. Percentages represent fractions of 100, making them easy to grasp and apply.
For instance, in the sale price example, if an item is 20% off, we're essentially taking 20 parts out of every 100 parts of the original price. This translates mathematically to multiplying the original price by 0.80.
Let's break down the calculation for a clearer understanding:

• Original Price (P) = \(50
• Discount = 20% of P = 0.20 * \)50 = \(10
• Sale Price (S) = P - Discount = \)50 - \(10 = \)40

Using the function notation we derived earlier:
$$ S = 0.80P $$
If P is \(50, then S is 0.80 * 50 = \)40.
Understanding percentages helps in recognizing how big or small a fractional part is in relation to the whole.

Time Conversion

Time conversion is vital when dealing with different time zones. In our example, Paris time (P) is 6 hours ahead of New York time (N).
The relationship between these times can be written as:
$$ P = N + 6 $$
For example, if it’s 3 PM in New York, in Paris it will be 9 PM. However, times are often expressed on a 12-hour clock.
If the calculated Paris time exceeds 12, we need to adjust it using the modulo operation. This means subtracting 12 from the result to fit it into the 12-hour clock format. The formula for this adjustment is:
$$ P = (N + 6) \bmod 12 $$
Using an example:

• New York Time (N) = 8 PM
• Paris Time without adjustment (P) = N + 6 = 14
• Paris Time adjusted, P = 14 mod 12 = 2 PM

This conversion ensures that time is represented correctly in a 12-hour system.

Wind Chill Temperature

Wind chill temperature illustrates how wind speed affects perceived temperature. When the air moves rapidly over exposed skin, it draws heat away, making it feel colder than the actual temperature.
To calculate the effective wind chill temperature (C), we need the actual outdoor temperature (T) and the wind speed (W). The formula expressing C as a function of T and W is:
$$ C = T - \frac{2}{3}W $$
Here’s a step-by-step breakdown:

• Actual Temperature (T) = 30°F
• Wind Speed (W) = 15 mph
• Wind Chill Effect = \(\frac{2}{3}\) of W = \(\frac{2}{3} \times 15\) = 10°F
• Effective Wind Chill Temperature (C) = T - Wind Chill Effect = 30°F - 10°F = 20°F

Using this formula helps predict how cold it will feel, which is crucial for preparing for outdoor activities and staying safe in cold weather.