Question

Consider the following.

Slick Slack is a downloading app that can be used to automatically generate excuses for being late to work based on the weather, traffic conditions, and the employer’s line of business. One week after its release, 6,250 copies of Slick Slack were downloaded. After five weeks, 13,700 copes had been downloaded.

Let x represent the number of weeks since slick slack was released, and y represent the number of copies that had been downloaded.

Use the marginal change and the fact that 1 week after it’s release 6,250 copies of Slick Slack were downloaded to determine the y-intercept in the equation y = a + bx.

(x, y) = _____

Write the equation in the form y = a + bc for the number of copies you downloaded x weeks after its release.

_______

Answer 1

• y-intercept: (0, 4387.5)
• y = 4387.5 +1862.5x

Explanation:

Given x represents weeks and y represents downloads, and you have the (x, y) values (1, 6250) and (5, 13700), you want the marginal change, the y-intercept, and the slope-intercept equation for the relation between x and y.

Marginal change

The marginal change, also called the slope, is the coefficient 'b' in your equation y = a +bx. Its value can be found by ...

b = (y2 -y1)/(x2 -x1)

b = (13700 -6250)/(5 -1) = 7460/4 = 1862.5

Y-intercept

The equation y = a +bx can be rearranged to give the value of 'a'.

a = y -bx

a = 6250 -(1862.5)(1) . . . . . . using (x, y) = (1, 6250)

a = 4387.5

The y-intercept in the equation is (x, y) = (0, 4387.5).

Equation

The equation for the number of copies downloaded is ...

y = a +bx

y = 4387.5 +1862.5x

__

Additional comment

The "marginal change" is the ratio of the change in output to the change in input. When the relation between output and input is linear, as here, the marginal change is the same for any magnitude of change.

When the relation is nonlinear, we are often interested in the limit of the marginal change as the changes from a given point become small. This limiting value is the derivative of the relation at that point.

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