To analyze the river's water level during the rain, let's break down the provided information step-by-step.
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Calculate the rate of change: The rate of change can be determined by finding the change in water level over the change in days.
- Water level after 3 days = 34 feet
- Water level after 10 days = 62 feet
- Change in water level = 62 - 34 = 28 feet
- Change in time = 10 - 3 = 7 days
Rate of change = Change in water level / Change in time = 28 feet / 7 days = 4 feet per day.
Answer: Rate of change = 4 feet per day.
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Find the y-intercept: The y-intercept occurs when time (x) is 0 (before the rain starts). It is given as 22 feet. Thus, the y-intercept is:
Answer: y-intercept = (0, 22).
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Write an equation in Point-slope form: Using the point (3, 34) and the rate of change (slope) of 4, we use the point-slope form of a linear equation, which is: \( y - y_1 = m(x - x_1) \) where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope.
Using (3, 34): \( y - 34 = 4(x - 3) \).
Answer: \( y - 34 = 4(x - 3) \).
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Change that equation to Standard form: From the point-slope form, we rearrange \( y - 34 = 4(x - 3) \): \[ y - 34 = 4x - 12 \] \[ y = 4x + 22 \] Rearranging to standard form \( Ax + By = C \) gives: \[ -4x + y = 22 \quad \Rightarrow \quad 4x - y = -22. \]
Answer: \( 4x - y = -22 \).
To summarize:
- Rate of change: 4 feet per day.
- Y-intercept: (0, 22).
- Point-slope form: \( y - 34 = 4(x - 3) \).
- Standard form: \( 4x - y = -22 \).