Question

average annual sales per worker of a firm is influenced by the number of hours, t, that employers spend training their workers in a year. the relationship is modeled by the function s(t)=85000 + 60000 ln(t). find the derivative s(t). then compute s(40). (a) s(t)= (b) s(40)= (round to two decimal places as needed.)

Explanation:

Step1: Recall derivative rules

The derivative of a constant $C$ is $0$, and the derivative of $\ln(t)$ is $\frac{1}{t}$. For the function $S(t)=85000 + 60000\ln(t)$, using the sum - rule of derivatives $\frac{d}{dt}(u + v)=\frac{du}{dt}+\frac{dv}{dt}$, where $u = 85000$ and $v=60000\ln(t)$.

Step2: Differentiate each term

The derivative of $85000$ with respect to $t$ is $0$, and the derivative of $60000\ln(t)$ with respect to $t$ is $60000\times\frac{1}{t}=\frac{60000}{t}$. So $S'(t)=\frac{60000}{t}$.

Step3: Evaluate $S'(40)$

Substitute $t = 40$ into $S'(t)$. We get $S'(40)=\frac{60000}{40}=1500.00$.

Answer:

(a) $S'(t)=\frac{60000}{t}$
(b) $S'(40)=1500.00$