a) The slope of the <em>secant</em> lines are: q = 5, f(q) = 2,076: m = - 131.1, q = 10, f(q) = 1,299: m = - 106.8, q = 20, f(q) = 357: m = - 81.6, q = 25, f(q) = 84: m = - 68.1, q = 30, f(q) = 0: m = - 51.
b) The <em>estimated</em> slope of the line <em>tangent to</em> the curve at the point (x, y) = (15, 765) is - 94.2 gallons per minute.
<h3>How to estimate the slope of a tangent line by averaging two adjacent secant lines</h3>
In this problem we must determine the slope of <em>several</em> lines based on the information given by the table and using the <em>secant line</em> formula:
m = [f(q) - f(p)] / (q - p) (1)
If we know that a = 15 and p = 765, then the slope of the secant lines are:
q = 5, f(q) = 2,076
m = [2,076 - 765] / (5 - 15)
m = - 131.1
q = 10, f(q) = 1,299
m = [1,299 - 765] / (10 - 15)
m = - 106.8
q = 20, f(q) = 357
m = [357 - 765] / (20 - 15)
m = - 81.6
q = 25, f(q) = 84
m = [84 - 765] / (25 - 15)
m = - 68.1
q = 30, f(q) = 0
m = [0 - 765] / (30 - 15)
m = - 51
The slope of the line <em>tangent</em> to the curve at the point (x, y) = (15, 765) can be estimated by averaging the slopes of the closest <em>secant</em> lines:
m' = [(- 106.8) + (- 81.6)] / 2
m' = - 94.2
The <em>estimated</em> slope of the line <em>tangent to</em> the curve at the point (x, y) = (15, 765) is - 94.2 gallons per minute.
To learn more on tangent lines: brainly.com/question/23265136
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