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Questions in mathematics

[Done] Rationalise i.) $\frac{8}{3 \sqrt{2}}$

[Done] Differentiate [tex]F^{\prime}(x)=x^3[/tex] from first principle

[Done] Solve for x. $\begin{array}{c} 3(2 x+4)=24 \\ x=[?] \end{array}$

[Done] In year 13, the scientist will put tree wrap around tree 1 to protect it from the winter snow. The height of the tree wrap needs to be 45 inches. The wrap is priced by the square foot. To the nearest square foot, how many square feet of wrap does she need? Tree 1 \begin{tabular}{|c|c|} \hline Year & \begin{tabular}{c} Trunk \\ Diameter \\ (inches) \end{tabular} \\ \hline 1 & 18.6 \\ \hline 3 & 19.2 \\ \hline 5 & 19.8 \\ \hline 7 & 20.4 \\ \hline 9 & 21.0 \\ \hline 11 & 21.6 \\ \hline 13 & 22.2 \\ \hline \end{tabular} Tree 2 \begin{tabular}{|c|c|} \hline Year & \begin{tabular}{c} Trunk \\ Diameter \\ (inches) \end{tabular} \\ \hline 1 & 11.4 \\ \hline 3 & 12.0 \\ \hline 5 & 12.6 \\ \hline 7 & 13.2 \\ \hline 9 & 13.8 \\ \hline 11 & 14.4 \\ \hline 13 & 15.0 \\ \hline \end{tabular}

[Done] Solve [tex]$\frac{1}{x^2}=\frac{1}{6 x^2}+\frac{1}{6 x}$[/tex].

[Done] You are standing next to a really big circular lake. You want to measure the diameter of the lake, but you don't want to have to swim across with a measuring tape! You decide to walk around the perimeter of the lake and measure its circumference, and find that it's [tex]$400 \pi m$[/tex]. What is the diameter [tex]$d$[/tex] of the lake? [tex]d=\square m$[/tex]

[Done] What is the vertex of the quadratic function [tex]$f(x)=(x-6)(x+2)$[/tex]?

[Done] Subtract these polynomials. $\left(6 x^2-x+8\right)-\left(x^2+2\right)=$ A. $7 x^2-x+10$ B. $5 x^2-x+10$ C. $5 x^2-x+6$ D. $7 x^2-x+6$

[Done] Multiply the polynomials. $(x+2)\left(x^2-7 x+4\right)$ A. $x^3-7 x^2-10 x+8$ B. $x^3-5 x^2-10 x+8$ C. $x^3-7 x^2+4 x+8$ D. $x^3-5 x^2+4 x+8$

[Done] Solve the quadratic equation $3 x^2+x-5=0$. Give your answers to 2 decimal places.