Question

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Explanation:

Step1: Identify Variables

Independent variable: time (\(x\)), Dependent variable: height (\(y\)). Ordered pairs: \((2, 110)\) and \((6, 30)\).

Step2: Calculate Slope

Slope \(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{30 - 110}{6 - 2} = \frac{-80}{4} = -20\).

Step3: Find Equation

Using point-slope form \(y - y_1 = m(x - x_1)\) with \((2, 110)\):
\(y - 110 = -20(x - 2)\)
\(y - 110 = -20x + 40\)
\(y = -20x + 150\).

Step4: Domain and Range

Domain: \(x \geq 0\) (time can’t be negative) until \(y = 0\): \(0 = -20x + 150 \Rightarrow x = 7.5\), so \(0 \leq x \leq 7.5\).
Range: \(y \geq 0\) (height non-negative) and \(y \leq 150\) (initial height? Wait, at \(x=0\), \(y=150\)? Wait, when \(x=2\), \(y=110\); \(x=6\), \(y=30\). Wait, maybe initial height is 150? Wait, let's check \(x=0\): \(y = -20(0) + 150 = 150\). So range: \(0 \leq y \leq 150\).

Step5: Slope Interpretation

Slope \(-20\) means the hot air balloon descends 20 meters per minute.

Step6: Intercepts

• \(x\)-intercept: Set \(y=0\): \(0 = -20x + 150 \Rightarrow x = 7.5\), so \((7.5, 0)\). This means at 7.5 minutes, the balloon reaches the ground.
• \(y\)-intercept: Set \(x=0\): \(y = 150\), so \((0, 150)\). This means at time 0 (start), the balloon is 150 meters high.

Answer:

• Independent: time (\(x\)), Dependent: height (\(y\))
• Ordered pairs: \((2, 110)\), \((6, 30)\)
• Equation: \(y = -20x + 150\)
• Domain: \(0 \leq x \leq 7.5\)
• Range: \(0 \leq y \leq 150\)
• Slope: \(-20\) (descends 20 m/min)
• \(x\)-intercept: \((7.5, 0)\) (time to land)
• \(y\)-intercept: \((0, 150)\) (initial height)