Problem 67 - Precalculus Student Solutions Manual 5th

Question

Temperature of Venus Venus is the hottest planet with a surface temperature of \(867^{\circ} \mathrm{F}\). What is this temperature in Celsius? (Source: World Almanac and Book of Facts.)

Step-by-Step Solution

Answer

Approximately \(463.89^{\u03bFC}\).

Step 1 - Recall the Conversion Formula

The formula to convert Fahrenheit to Celsius is given by \(C = \frac{5}{9} (F - 32)\), where \(C\) is the temperature in Celsius, and \(F\) is the temperature in Fahrenheit.

Step 2 - Substitute the Given Value

Substitute the given Fahrenheit temperature (\(867^{\circ} \text{F}\)) into the formula:\[C = \frac{5}{9} (867 - 32)\]

Step 3 - Simplify the Expression

First, compute the value inside the parentheses:\(867 - 32 = 835\).Next, multiply by \(\frac{5}{9}\):\[C = \frac{5}{9} \times 835\]

Step 4 - Perform the Multiplication

Compute the multiplication:\[C = \frac{5 \times 835}{9} = \frac{4175}{9} \approx 463.89\]

Step 5 - State the Final Answer

The temperature of Venus in Celsius is approximately \(463.89^{\circ} \text{C}\).

Key Concepts

Fahrenheit to Celsius Conversion

Converting Fahrenheit to Celsius involves using a specific formula. This formula is: \(C = \frac{5}{9} (F - 32)\). Here, \(C\) represents the temperature in Celsius and \(F\) represents the temperature in Fahrenheit.
To make the conversion, you need to follow these steps:

• Subtract 32 from the Fahrenheit temperature. This shifts the scale down to align with zero on the Celsius scale.
• Multiply the result by \(\frac{5}{9}\). This adjusts the scale based on the difference in size between Celsius and Fahrenheit degrees. Fahrenheit degrees are smaller, so the multiplication factors this difference in.

By following these steps, you can convert any Fahrenheit temperature to Celsius.

Surface Temperature

Surface temperature refers to the temperature at the surface of a planet. For instance, the surface temperature of Venus is a scorching \(867^{\circ} \text{F}\). Surface temperature can be measured using various tools like satellites and ground-based sensors.
Studying surface temperature is important because it helps scientists understand the climate and atmospheric conditions of a planet. For example, Venus has a high surface temperature due to its thick atmosphere filled with carbon dioxide, which traps heat.
When learning about different planets, knowing their surface temperatures can give insights into their environments and potential for supporting life.

Multiplication

Multiplication is a basic mathematical operation that can be seen in various calculations, including temperature conversions. In the example of converting \(867^{\circ} \text{F}\) to Celsius, multiplication occurs when you apply the factor \(\frac{5}{9}\). Here’s a step-by-step on how multiplication plays a role:

• After subtracting 32 from 867, you get 835.
• Next, multiply 835 by \(\frac{5}{9}\). This can be broken down into a two-step operation: \(835 \times 5 = 4175\), and then \(4175 \div 9 \).
• The outcome (approx. 463.89) represents the temperature in Celsius.

Multiplication helps rescale values and is essential in many scientific and everyday calculations.