The Distance from the top of a cliff to the water below ranges from 13m(low tide) to 8m(high tide). Low tide occurs at 12:00 noon and it takes six hours between low tide and high tide.
a)Develop an equation representing the distance in terms of time
b) It is only safe to dive into the water when the distance is 10m or less, between what times is it safe to dive? round to the nearest minute
a) To develop an equation representing the distance in terms of time, we can use the information given.
We know that it takes six hours between low tide and high tide. Low tide occurs at 12:00 noon, so high tide would occur six hours later, at 6:00 PM.
Let's call the time elapsed since low tide as "t" hours. At low tide, the distance from the top of the cliff to the water below is 13 meters, and at high tide, it is 8 meters.
Since the distance decreases linearly as time passes, we can use the slope-intercept form of a linear equation: y = mx + b. In this case, "y" represents the distance and "x" represents time.
To find the slope, we need to calculate the change in distance between high and low tide, and divide it by the change in time:
slope = (8 - 13) / (6 - 0)
= -5 / 6
Now we have our slope. We also have a point on the line, (0, 13), as it represents the distance at low tide. We can substitute these values into the equation to find the y-intercept (b):
13 = (-5 / 6)(0) + b
b = 13
Therefore, the equation representing the distance in terms of time is:
distance = (-5/6) * time + 13
b) To determine the times when it is safe to dive (when the distance is 10 meters or less), we can substitute 10 for "distance" in the equation:
10 = (-5/6) * time + 13
Now we can solve for "time":
-5/6 * time = 10 - 13
-5/6 * time = -3
To isolate "time," we can multiply both sides by -6/5 to cancel out the coefficient:
time = (-3) * (-6/5)
time = 18/5
time = 3.6 hours
To convert this to minutes, we multiply by 60:
time = 3.6 * 60
time = 216 minutes
Therefore, it is safe to dive between approximately 12:00 noon and 3:26 PM (rounded to the nearest minute).