Questions in mathematics

Questions in mathematics

📝 Answered - A tool manufacturer wants to produce a wrench set that consists of a toolbox and 7 wrenches. The toolbox costs $t$ dollars to produce and each wrench costs $w$ dollars to produce. The total cost of producing the wrench set is at least 4 times but no more than 6 times the cost of producing the toolbox. Which inequality represents all possible values of [tex]$w$[/tex]? (A) [tex]$\frac{3}{7} t \leq w \leq \frac{5}{7} t$[/tex] (B) [tex]$3 t \leq w \leq 5 t$[/tex] (C) [tex]$\frac{4}{7} t \leq w \leq \frac{6}{7} t$[/tex] (D) [tex]$4 t \leq w \leq 6 t$[/tex]

📝 Answered - Show that the point of inflection of [tex]f(x)=x(x-21)^2[/tex] lies midway between the relative extrema of [tex]f[/tex]. Begin by finding the first derivative. [tex]f^{\prime}=\square[/tex] Next, find the second derivative. [tex]f^*=\square[/tex] Then, find the relative extrema. [tex] \begin{array}{ll} \text { smaller } x \text {-value } & (x, y)=(\square) \\ \text { larger } x \text {-value } & (x, y)=(\square) \end{array} [/tex] Finally, find the inflection point. [tex](x, y)=(\square)[/tex] This confirms that the inflection point is midway between the relative extrema of [tex]f[/tex].

📝 Answered - Solve the equation. [tex]\begin{array}{c} \sqrt{3 x-4}=2 \sqrt{x-5} \\ x=[?]\end{array}[/tex]

📝 Answered - (1) If [tex]$f(x)=\sqrt{2 x-1}$[/tex] and [tex]$g(x)=1 / x$[/tex], find: (a) [tex]$\left(\frac{f}{g}\right)(x)$[/tex] (b) [tex]$(f+g)(x)$[/tex]

📝 Answered - What values of $a$ and $b$ make the equation true? [tex]\sqrt{648}=\sqrt{2^a \cdot 3^b}[/tex] A. $a=3, b=2$ B. $a=2, b=3$ C. $a=3, b=4$ D. $a=4, b=3$

📝 Answered - Rewrite without rational exponents, and simplify, if possible. $81^{\frac{3}{4}}$

📝 Answered - Solve. [tex]$\sqrt{8 x-13}=1$[/tex] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution is [tex]$x=$[/tex] $\square$ (Simplify your answer. Type an integer or a fraction.) B. There is no solution.

📝 Answered - Used books cost $2.25 each. Which expression represents the total number of books that can be purchased for $10? [tex]$\left\lfloor\frac{10}{2.25}\right\rfloor$[/tex] [tex]$\left\lceil\frac{10}{2.25}\right\rceil$[/tex] [tex]$\frac{10}{\lceil 2.25\rceil}$[/tex] [tex]$\frac{10}{\lfloor 2.25\rfloor}$[/tex]

📝 Answered - Let $G$ be a commutative group, and let $g \in G$ be an element of maximal finite order, that is, such that if $h \in G$ has finite order, then $|h| \leq|g|$. Prove that in fact $h$ has finite order in $G$, then $|h|$ divides $|g|$. (Hint: Argue by contradiction. If $|h|$ is finite but does not divide $|g|$, then there is a prime integer $p$ such that $|g|=p^r m,|h|=p^s n$, with $r$ and $s$ are relatively prime to $p$ and $m

📝 Answered - Which equation has the least steep graph? A. $y=-\frac{1}{2} x-3$ B. $y=-6 x+1$ C. $y=2 x-7$ D. $y=\frac{1}{4} x+9$