Rose Harrison, Clara Huizink, Aidan Sproat Clements and, Marlene Torres Skoumal Solutions for Exercise 18: Mixed practice

Solve algebraically and confirm graphically,

$2x^2-4x<7x$

Solve algebraically and confirm graphically,

$5x^2-13x\ge -6$

Solve algebraically and confirm graphically,

$\frac{x-3}{x+5}>0, x\ne -5$

Solve algebraically and confirm graphically,

$\frac{3x-5}{x-1}>4, x\ne 1$

A rectangular playing field with a perimeter of $100 \mathrm{m}$ is to have an area of no less than $500 {\mathrm{m}}^2$. Determine the maximum possible length of the playing field.

A travel agency's profits $P$ (in dollars) can be modelled by the function $P\left(x\right)=-25x^2+1000x-3000$, where $x$ is the number of tourists. Determine the range of the number of tourists needed for the agency's profit to be at least $\$5000$.

Orange juice is to be sold in $2$-liter cylindrical cans. To keep costs down, the surface area of a can must be less than $1000 {\mathrm{cm}}^2$. Determine possible values for the radius and height of the cans, accurate to $1$ decimal point. (A liter is $1000 {\mathrm{cm}}^3$)

A packing company designs boxes with a volume of at least $600$ cubic inches. Squares are cut from the corners of a $20$ inch by $25$ inch rectangle of cardboard, and the flaps are folded up to make an open box. Determine the size of the squares that should be cut from the cardboard, accurate to $3$ significant figures.