The problem presented involves creating a mathematical equation based on a scenario where scientists are studying the temperature on a distant planet. Here is the breakdown of the information provided and how to create the equation:

1. Given Information:

   • The surface temperature on the planet (at $H = 0$, where $H$ is the height above the surface in kilometers) is $30^\circ$ Celsius.
   • The temperature decreases by $3^\circ$ Celsius for each kilometer you go up from the surface.

2. Define Variables:

   • Let $T$ represent the temperature (in Celsius) at height $H$ kilometers above the surface.
   • $H$ is the height in kilometers.

3. Creating the Equation:

   • Since the temperature decreases linearly with height, we can use a linear equation to model this situation.
   • The rate of change (slope) of the temperature with respect to height is $-3$ (since temperature decreases by $3^\circ$ per kilometer).
   • The initial value (y-intercept) is $30$ (the temperature at $H = 0$).

   Using the slope-intercept form of a line $T = mH + b$, where $m$ is the slope and $b$ is the y-intercept, we get:

   $T = -3H + 30$

4. Final Equation:

   • The equation that models the temperature $T$ in terms of the height $H$ is: $T = -3H + 30$

This equation indicates that for every 1 kilometer increase in height ($H$), the temperature $T$ decreases by 3 degrees Celsius.

Would you like me to go over the process in more detail or provide guidance on graphing this equation?

Related Questions

1. What would the temperature be at a height of 5 kilometers?
2. How would the equation change if the temperature decreased by 5 degrees per kilometer instead?
3. Can you explain why this is a linear relationship rather than a different type of function?
4. How would you determine the height at which the temperature reaches 0 degrees Celsius?
5. What would the equation look like if the initial temperature at the surface was different, say 40 degrees Celsius?

Tip

When creating linear equations based on word problems, identifying the initial value and rate of change can quickly help set up the equation in slope-intercept form.