A record label uses the following function to model the sales of a new release. a (t) = - 90t 2 + 8100t The number of albums sold is a function of time, t, in days. On which day were the most albums sold? What is the maximum number of albums sold on that day?

Question

Mathematics

Kailyn Bolton

1 October, 12:24

A record label uses the following function to model the sales of a new release. a (t) = - 90t 2 + 8100t The number of albums sold is a function of time, t, in days. On which day were the most albums sold? What is the maximum number of albums sold on that day?

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Reed Campbell · Answer 1

Most albums were sold on day 45th after the release.

The number of albums sold on that day is 182250.

Step-by-step explanation:

Hi there!

Let's write the function:

a (t) = - 90t² + 8100t

Where "a" is the number of albums sold and "t" is the time in days from the release.

To find the maximum of a function, we have to find the value of "t" at which the line tangent to the function is horizontal, that is, its slope is zero. Besides, the function must increase at the "t" values to the left of the maximum and decrease at the "t" values located to the right of the maximum. That means that the slopes of the lines tangent to the function at the left of the maximum must be positive, while the slopes of the tangent lines must be negative at the right of maximum (see figure for a better understanding)

Then let's find the value of t for which the slope of the tangent line is zero. Be have to find the derivative of a (t), a' (t):

a (t) = - 90t² + 8100t

a' (t) = 2 · (-90t) + 8100

Now we have to find the value of "t" for which the derivative is zero:

0 = - 180t + 8100

-8100 = - 180t

-8100/-180 = t

t = 45

We have found that at t = 45 the slope of the tangent line is zero. This can be a maximum, a minimum or just a place where the function is horizontal (the function is a negative parabola, so we can be sure that this is a maximum but let's corroborate it).

Let's find the slope of tangents lines at t values to the left and right of t = 45. For example, let's take 44.9 and 45.1:

a' (t) = - 180t + 8100

a' (44.9) = - 180 (44.9) + 8100 = 18 (the slope is positive, the function increases to the left of t = 45).

a' (45.1) = - 180 (45.1) + 8100 = - 18 (the slope is negative, the function decreases to the right of t = 45)

Then there is a maximum at t = 45 days after the release.

To find the maximum number of albums sold on that day, we have to evaluate the function a (t) at t = 45.

a (t) = - 90t² + 8100t

a (45) = - 90· (45) ² + 8100· (45) = 182250

On that day, the number of albums sold is 182250.