Scientists are studying the temperature on a distant planet. Let y represent the temperature (in degrees Celsius). Let x represent the height above the surface (in kilometers). Suppose that x and y are related by the equation .
41-5x=y
Answer the questions below.
Note that a change can be an increase or a decrease.
For an increase, use a positive number. For a decrease, use a negative number.
What is the temperature on the surface of the planet?
What is the temperature change for each kilometer we go up from the surface?
To find the temperature on the surface of the planet, we can substitute x = 0 into the equation 41-5x=y:
41 - 5(0) = y
41 = y
Therefore, the temperature on the surface of the planet is 41 degrees Celsius.
To find the temperature change for each kilometer we go up from the surface, we can calculate the difference in temperatures between successive heights by subtracting the temperatures at those heights.
Let's consider two heights, x1 and x2, such that x2 is 1 kilometer above x1. The temperatures at these heights can be calculated using the equation 41-5x=y:
Temperature at x1: 41-5x1 = y1
Temperature at x2: 41-5x2 = y2
The temperature change for each kilometer we go up from the surface, Δy, can be calculated as:
Δy = y2 - y1 = (41-5x2) - (41-5x1)
Simplifying this expression gives:
Δy = 41 - 5x2 - 41 + 5x1
Since x2 is 1 kilometer above x1, we can substitute x2 = x1 + 1 into the expression:
Δy = 41 - 5(x1 + 1) - 41 + 5x1
= 41 - 5x1 - 5 - 41 + 5x1
= -5
Therefore, the temperature change for each kilometer we go up from the surface is -5 degrees Celsius. This means that for each kilometer increase in height above the surface, the temperature decreases by 5 degrees Celsius.