Question

a store is having a 14 hour sale. the total number of shoppers that have entered the store after t hours from the beginning of the sale is modeled by the function (\frac{1}{2}t^4 - 6t^3 + 86t^2). at what rate are the shoppers entering the store at (t = 5) hours? (2t^3 - 18t^2 + 172t)

Explanation:

Step1: Identify rate function (derivative)

The rate of shoppers entering is the derivative of the total shoppers function. Given the derivative is provided as:
$2t^3 - 18t^2 + 172t$

Step2: Substitute $t=5$ into rate function

Replace $t$ with 5 in the polynomial:
$2(5)^3 - 18(5)^2 + 172(5)$

Step3: Calculate each term separately

$2(125) - 18(25) + 172(5) = 250 - 450 + 860$

Step4: Compute final value

Add and subtract the terms:
$250 - 450 + 860 = 660$

Answer:

660 shoppers per hour