Questions: The total sales of a company (in millions of dollars) t months from now are given by the following formula. S(t)=3 sqrt(t+1) (A) Use the four-step process to find S'(t). (B) Find S(15) and S'(15). (C) Use the results in part (B) to estimate the total sales after 16 months and 17 months. (A) S'(t)= (B) S(15)= (Type an integer or a decimal.) S'(15)= (Type an integer or a decimal.) (C) S(16) ≈ (Type an integer or a decimal.) S(17) ≈ (Type an integer or a decimal.)

The total sales of a company (in millions of dollars) t months from now are given by the following formula.
S(t)=3 sqrt(t+1)
(A) Use the four-step process to find S'(t).
(B) Find S(15) and S'(15).
(C) Use the results in part (B) to estimate the total sales after 16 months and 17 months.
(A) S'(t)=
(B) S(15)= (Type an integer or a decimal.)
S'(15)= (Type an integer or a decimal.)
(C) S(16) ≈ (Type an integer or a decimal.)
S(17) ≈ (Type an integer or a decimal.)

Transcript text: The total sales of a company (in millions of dollars) t months from now are given by the following formula. \[ S(t)=3 \sqrt{t+1} \] (A) Use the four-step process to find $\mathrm{S}^{\prime}(\mathrm{t})$. (B) Find $S$ (15) and $S^{\prime}(15)$. (C) Use the results in part (B) to estimate the total sales after 16 months and 17 months. (A) $\mathrm{S}^{\prime}(\mathrm{t})=$ $\square$ (B) $\mathrm{S}(15)=$ $\square$ (Type an integer or a decimal.) $S^{\prime}(15)=$ $\square$ (Type an integer or a decimal.) (C) $\mathrm{S}(16) \approx$ $\square$ (Type an integer or a decimal.) $\mathrm{S}(17) \approx$ $\square$ (Type an integer or a decimal.)

Solution

Solution Steps

Solution Approach

(A) To find $ S'(t) $, we need to differentiate the function $ S(t) = 3\sqrt{t+1} $ with respect to $ t $. Use the chain rule for differentiation.

(B) To find $ S(15) $, substitute $ t = 15 $ into the function $ S(t) $. To find $ S'(15) $, substitute $ t = 15 $ into the derivative $ S'(t) $.

(C) Use the linear approximation formula $ S(t+1) \approx S(t) + S'(t) \cdot 1 $ to estimate the sales after 16 and 17 months using the results from part (B).

Step 1: Find $ S'(t) $

To find the derivative of the sales function $ S(t) = 3\sqrt{t+1} $, we apply the chain rule. The derivative is given by: \[ S'(t) = \frac{3}{2\sqrt{t+1}} \]

Step 2: Calculate $ S(15) $ and $ S'(15) $

Substituting $ t = 15 $ into the sales function: \[ S(15) = 3\sqrt{15+1} = 3\sqrt{16} = 12 \] Now, substituting $ t = 15 $ into the derivative: \[ S'(15) = \frac{3}{2\sqrt{15+1}} = \frac{3}{2\sqrt{16}} = \frac{3}{8} \]

Step 3: Estimate $ S(16) $ and $ S(17) $

Using the linear approximation formula: \[ S(16) \approx S(15) + S'(15) \cdot 1 = 12 + \frac{3}{8} = \frac{99}{8} \] \[ S(17) \approx S(15) + S'(15) \cdot 2 = 12 + 2 \cdot \frac{3}{8} = 12 + \frac{6}{8} = \frac{51}{4} \]

Final Answer

\[ S'(t) = \frac{3}{2\sqrt{t+1}}, \quad S(15) = 12, \quad S'(15) = \frac{3}{8}, \quad S(16) \approx \frac{99}{8}, \quad S(17) \approx \frac{51}{4} \] Thus, the final boxed answers are: \[ \boxed{S'(t) = \frac{3}{2\sqrt{t+1}}} \] \[ \boxed{S(15) = 12} \] \[ \boxed{S'(15) = \frac{3}{8}} \] \[ \boxed{S(16) \approx \frac{99}{8}} \] \[ \boxed{S(17) \approx \frac{51}{4}} \]

Was this solution helpful?

Unhelpful

Helpful

Questions asked by other users

< Previous Next >

Thank you for your feedback.

Copied!