Question

Solve the problem.When exposed to ethylene gas, green bananas will ripen at an accelerated rate. The number of days for ripening becomes shorter for longer exposure times. Assume that the table below gives average ripening times of bananas for several different ethylene exposure times. Plot the data and then find a line approximating the data. With the aid of this line, determine the rate of change of ripening time with respect to the exposure time. Round your answer to two significant digits.

Answer 1

The table and the plotted data are in the attachment.

Answer and Step-by-step explanation: A line of best fit is the best approximation of the given data. It is used to show the relation of two variables and it can be determined, more accurately, by the Least Square Method.

To use the method, follow the steps:

1) Calculate the mean of the two variables:

$X = \frac{\Sigma x_{i}}{n}$

$Y=\frac{\Sigma y_{i}}{n}$

2) The Slope of the line is given by:

$m=\frac{\Sigma (x_{i}-X)(y_{i}-Y)}{\Sigma (x_{i}-X)^{2}}$

3) The y-intercept is found by the formula:

$b=Y-mX$

For the ripening of bananas, the least square method gives the line:

y = -0.142x + 5.56

The rate of change of a line equation is its slope, so

Rate of change = -0.14

The rate of change of ripening time with respect to exposure time is -0.14, which means the more expose the fruit is, the less time it needs to ripe.

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