attempt 1: 10 attempts remaining. the profit, in dollars, from selling x number of magazines is given by…
attempt 1: 10 attempts remaining. the profit, in dollars, from selling x number of magazines is given by: $p(x)=4x+sqrt{7x + 26}$. find the rate of change of profit when 210 magazines are sold. (round your answer to the nearest dollar.) the profit is? by? submit answer next item
Answer
Explanation:
Step1: Find the derivative of $P(x)$
The derivative of $4x$ is $4$ using the power - rule ($\frac{d}{dx}(ax)=a$ for a constant $a$). For $y = \sqrt{7x + 26}=(7x + 26)^{\frac{1}{2}}$, by the chain - rule $\frac{dy}{dx}=\frac{1}{2}(7x + 26)^{-\frac{1}{2}}\times7=\frac{7}{2\sqrt{7x + 26}}$. So, $P^\prime(x)=4+\frac{7}{2\sqrt{7x + 26}}$.
Step2: Evaluate $P^\prime(x)$ at $x = 210$
Substitute $x = 210$ into $P^\prime(x)$: [ \begin{align*} P^\prime(210)&=4+\frac{7}{2\sqrt{7\times210+26}}\ &=4+\frac{7}{2\sqrt{1470 + 26}}\ &=4+\frac{7}{2\sqrt{1496}}\ &=4+\frac{7}{2\times38.68}\ &=4+\frac{7}{77.36}\ &=4 + 0.09\ &\approx4 \end{align*} ]
Answer:
The profit is increasing by $4$ dollars.