Questions: 4) A company's total monthly sales (in millions of dollars) t months from now are given by: S(t)=0.5 t^2+3 t-5 a) Find S'(t) b) Find S(4) c) What does the value found in b) represent? d) Find S'(4) e) What does the value found in d) represent?

a) Find \( S^{\prime}(t) \)

Differentiate \( S(t) \) with respect to \( t \)

Given \( S(t) = 0.5t^2 + 3t - 5 \), the derivative \( S^{\prime}(t) \) is calculated as:
\[ S^{\prime}(t) = \frac{d}{dt}(0.5t^2) + \frac{d}{dt}(3t) + \frac{d}{dt}(-5) = t + 3 \]

\(\boxed{S^{\prime}(t) = t + 3}\)

b) Find \( S(4) \)

Substitute \( t = 4 \) into \( S(t) \)

\[ S(4) = 0.5(4)^2 + 3(4) - 5 = 0.5(16) + 12 - 5 = 8 + 12 - 5 = 15 \]

\(\boxed{S(4) = 15}\)

c) What does the value found in b) represent?

Interpret the meaning of \( S(4) \)

The value \( S(4) = 15 \) represents the company's total monthly sales (in millions of dollars) 4 months from now.

\(\boxed{\text{The company's total monthly sales 4 months from now is \$15 million.}}\)

d) Find \( S^{\prime}(4) \)

Substitute \( t = 4 \) into \( S^{\prime}(t) \)

\[ S^{\prime}(4) = 4 + 3 = 7 \]

\(\boxed{S^{\prime}(4) = 7}\)

e) What does the value found in d) represent?

Interpret the meaning of \( S^{\prime}(4) \)

The value \( S^{\prime}(4) = 7 \) represents the rate of change of the company's total monthly sales (in millions of dollars per month) 4 months from now.

\(\boxed{\text{The rate of change of the company's total monthly sales 4 months from now is \$7 million per month.}}\)

\(\boxed{S^{\prime}(t) = t + 3}\)
\(\boxed{S(4) = 15}\)
\(\boxed{\text{The company's total monthly sales 4 months from now is \$15 million.}}\)
\(\boxed{S^{\prime}(4) = 7}\)
\(\boxed{\text{The rate of change of the company's total monthly sales 4 months from now is \$7 million per month.}}\)