Question

attempt 1: 10 attempts remaining. the total ticket sales $s(x)$, in dollars, from selling $x$ tickets is modeled by: $s(x)=4x^{2}+sqrt{16x + 48}$. find the rate of change of ticket sales when 260 tickets are sold. (round your answer to the nearest dollar.) a) $s(260)=$ b) the ticket sales is by

Explanation:

Step1: Differentiate $4x^{2}$

Using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, for $y = 4x^{2}$, $y^\prime=8x$.

Step2: Differentiate $\sqrt{16x + 48}=(16x + 48)^{\frac{1}{2}}$

Using the chain - rule $\frac{d}{dx}(u^{n})=nu^{n - 1}\cdot u^\prime$. Let $u = 16x+48$, then $u^\prime=16$. So $\frac{d}{dx}(16x + 48)^{\frac{1}{2}}=\frac{1}{2}(16x + 48)^{-\frac{1}{2}}\cdot16=\frac{8}{\sqrt{16x + 48}}$.

Step3: Find $S^\prime(x)$

$S^\prime(x)=8x+\frac{8}{\sqrt{16x + 48}}$.

Step4: Evaluate $S^\prime(260)$

Substitute $x = 260$ into $S^\prime(x)$:
\[

$$\begin{align*} S^\prime(260)&=8\times260+\frac{8}{\sqrt{16\times260 + 48}}\\ &=2080+\frac{8}{\sqrt{4160+48}}\\ &=2080+\frac{8}{\sqrt{4208}}\\ &=2080+\frac{8}{64.87}\\ &\approx2080 + 0.12\\ &\approx2080 \end{align*}$$

\]

Answer:

a) $S^\prime(260)=2080$
b) The ticket sales is increasing by 2080 dollars.