Question

1. at the coast (elevation = 0 ft.), the temperature is 70°f. at the top of a nearby mountain (elevation = 10,000 ft.), the temperature is 30°f. assume a linear decrease in temperature with increasing elevation, as shown in the graph.

a. what is the slope of the line?
b. what is the y - intercept?
c. specify the domain and range of the graph. 0 <= x <= 10000, 30 <= y <= 70
d. what would the expected temperature be at 7,500 ft.?

Explanation:

Step1: Find slope formula

The slope $m$ of a line is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1,y_1)$ and $(x_2,y_2)$ are two points on the line. Here, $(x_1,y_1)=(0,70)$ (coast - elevation 0 ft, temp 70°F) and $(x_2,y_2)=(10000,30)$ (mountain - elevation 10000 ft, temp 30°F).

Step2: Calculate slope

$m=\frac{30 - 70}{10000-0}=\frac{- 40}{10000}=-0.004$.

Step3: Identify y - intercept

The y - intercept is the value of y when x = 0. When elevation $x = 0$ (coast), temperature $y = 70$, so the y - intercept is 70.

Step4: Determine domain and range

The domain is the set of all possible x - values (elevations). From the problem, $0\leq x\leq10000$. The range is the set of all possible y - values (temperatures), so $30\leq y\leq70$.

Step5: Find temperature at 7500 ft

The equation of the line is $y=mx + b$, where $m=-0.004$ and $b = 70$. Substitute $x = 7500$ into the equation: $y=-0.004\times7500+70$. First, calculate $-0.004\times7500=-30$. Then $y=-30 + 70=40$.

Answer:

a. The slope of the line is - 0.004.
b. The y - intercept is 70.
c. Domain: $0\leq x\leq10000$, Range: $30\leq y\leq70$.
d. The expected temperature at 7500 ft is 40°F.