Let h(12) = -100 for some function h(x), defined everywhere. If the average rate of change of h(x) from x = 12 to x = 20 is 50, then find h(20).
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To find h(20), we can use the given information about the average rate of change of h(x) from x = 12 to x = 20.
The average rate of change of a function h(x) over an interval [a, b] can be calculated using the formula:
Average rate of change = (h(b) - h(a)) / (b - a)
In this case, we have h(12) = -100 and the average rate of change = 50. We want to find h(20), so our interval is [12, 20] and a = 12, b = 20. Substituting these values into the formula, we get:
50 = (h(20) - (-100)) / (20 - 12)
Simplifying the equation:
50 = (h(20) + 100) / 8
Multiplying both sides by 8:
400 = h(20) + 100
Subtracting 100 from both sides:
h(20) = 300
Therefore, h(20) = 300.