Question

the total sales of a company (in millions of dollars) t months from now are given by the following formula. s(t)=3√(t + 10) (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.)

Explanation:

Step1: Rewrite the function

Rewrite $S(t)=3\sqrt{t + 10}=3(t + 10)^{\frac{1}{2}}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = ax^n$, then $y^\prime=anx^{n - 1}$. For $S(t)=3(t + 10)^{\frac{1}{2}}$, let $u=t + 10$, then $S(t)=3u^{\frac{1}{2}}$. By the chain - rule $\frac{dS}{dt}=\frac{dS}{du}\cdot\frac{du}{dt}$. $\frac{dS}{du}=3\times\frac{1}{2}u^{-\frac{1}{2}}$ and $\frac{du}{dt}=1$. So $S^\prime(t)=\frac{3}{2\sqrt{t + 10}}$.

Step3: Find $S(15)$

Substitute $t = 15$ into $S(t)$: $S(15)=3\sqrt{15 + 10}=3\sqrt{25}=3\times5 = 15$.

Step4: Find $S^\prime(15)$

Substitute $t = 15$ into $S^\prime(t)$: $S^\prime(15)=\frac{3}{2\sqrt{15+10}}=\frac{3}{2\sqrt{25}}=\frac{3}{10}=0.3$.

Step5: Estimate $S(16)$

Use the linear approximation formula $S(x)\approx S(a)+S^\prime(a)(x - a)$. Here $a = 15$ and $x = 16$. So $S(16)\approx S(15)+S^\prime(15)(16 - 15)=15+0.3\times1 = 15.3$.

Step6: Estimate $S(17)$

Using the linear approximation formula with $a = 15$ and $x = 17$, $S(17)\approx S(15)+S^\prime(15)(17 - 15)=15+0.3\times2 = 15.6$.

Answer:

(A) $S^\prime(t)=\frac{3}{2\sqrt{t + 10}}$
(B) $S(15)=15$, $S^\prime(15)=0.3$
(C) $S(16)\approx15.3$, $S(17)\approx15.6$