QuestionFind the rate of change of sales $S(t)=10,000+2000 t-200 t^{2}$ and interpret $S^{\prime}(t)$ .

Studdy Solution

STEP 1

Assumptions1. The sales volume $(t)$ is given by the function $(t)=10,000+2000t-200t^{}$ . $t$ is measured in weeks and $is the number of records sold per week3. We need to find the rate at which$ is changing, which is the derivative $'(t)$

STEP 2

To find the rate at which $$ is changing, we need to find the derivative of $(t)$ with respect to $t$. This is done using the power rule for differentiation, which states that the derivative of $x^n$ is $nx^{n-1}$.
$'(t) = \frac{d}{dt}(10,000+2000t-200t^{2})$

STEP 3

Now, differentiate each term of the function separately. The derivative of a constant is zero, the derivative of $t$ is1, and the derivative of $t^{2}$ is $2t$ .
$'(t) = \frac{d}{dt}(10,000)+\frac{d}{dt}(2000t)-\frac{d}{dt}(200t^{2})$

STEP 4

Calculate the derivative.
$'(t) =0 +2000 -400t$

STEP 5

implify the derivative.
$'(t) =2000 -400t$ This is the rate at which $$ is changing at any time $t$.

STEP 6

To interpret $'(t)$ , we can say that it represents the rate of change of the number of records sold per week with respect to time. If $'(t)$ is positive, the sales volume is increasing; if $'(t)$ is negative, the sales volume is decreasing. The magnitude of $'(t)$ gives the speed of this change.

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