BEquation: y=- 3.5x Interpretation of Slope: The temperature decreases 3.5^(∘) F for every 10 thousand feet.
C54.75 ^(∘) F
Practice makes perfect
The given graph relates the temperature change y in degrees Fahrenheit to the altitude change x in thousands of feet.
We want to know if x and y are in a proportional relationship. To do so, remember that when two quantities are proportional, the relationship can be represented by the following equation.
y=mx
Here, m is the constant of proportionality. The graph of this equation is a line with a slope m that passes through the origin. Notice that the given line passes through the origin. This means that x and y are in a proportional relationship.
Let's recall again the equation that describes a proportional relationship.
y=mx
To find the equation of the line, we will use the Slope Formula to obtain m.
m = y_2-y_1/x_2-x_1
Consider two points of the given graph to substitute into the Slope Formula. We will use the points ( 0, 0) and ( 10, - 35). Be aware that each value of the x-coordinate represents thousands of feet.
This means that the temperature decreases 3.5^(∘) F for every 10 thousand feet. Finally, we can substitute the slope into the equation of the proportional relationship.
y=- 3.5x
We know that the temperature at the bottom of a mountain is 74^(∘) F. We want to find the temperature at the top of the mountain if the top is 5500 feet above us. To do so, we can use the equation obtained in Part B.
y=- 3.5x
If we consider that the bottom has 0 altitude, the altitude change will be equal to the difference between 5500 feet and 0. Remember that the slope was obtained considering that each x-coordinate represents thousands of feet. Then, we can rewrite 5500 feet as 5.5 thousands of feet.
x=5.5-0=5.5
Now that we have the altitude change, we will substitute it into the equation obtained in Part B. Let's do it!
The temperature change is equal to 19.25 ^(∘) F. Finally, we can calculate the temperature at the top of the mountain by subtracting 19.25 ^(∘) F from the temperature at the bottom of the mountain.
74^(∘) -19.25^(∘) =54.75^(∘)
