Answer:
The highest price the bakery can charge, in dollars is $0.86623
Step-by-step explanation:
We are given profit function as
[tex]y=-1000x^2+110x-2.5[/tex]
where x is the price of a bagel in dollars
we are given least profit =200
so, we get inequality as
[tex]-1000x^2+1100x-2.5\geq 200[/tex]
now, we have to find highest value of price or x
So, we will solve for inequality and then we choose largest value of x
Firstly, we set it equal and then we solve for x
[tex]-1000x^2+1100x-2.5=200[/tex]
We will multiply both sides by 10
[tex]-1000x^2\cdot \:10+1100x\cdot \:10-2.5\cdot \:10=200\cdot \:10[/tex]
[tex]-10000x^2+11000x-25=2000[/tex]
[tex]-10000x^2+11000x-2025=0[/tex]
we can use quadratic formula
[tex]x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
now, we can compare and find a,b and c
a=-10000 , b=11000, c=-2025
we get
[tex]x=\frac{-11000+\sqrt{11000^2-4\left(-10000\right)\left(-2025\right)}}{2\left(-10000\right)}[/tex]
[tex]x=\frac{-11000-\sqrt{11000^2-4\left(-10000\right)\left(-2025\right)}}{2\left(-10000\right)}[/tex]
[tex]x=\frac{11-2\sqrt{10}}{20},\:x=\frac{11+2\sqrt{10}}{20}[/tex]
[tex]x=0.2337,x=0.86623[/tex]
Firstly, we will draw a number line and locate these values
and then we can check sign of inequality on each intervals
so, we got interval as
[tex]x:[0.2337,0.86623][/tex]
now, we can find largest x-value
x=0.86623
So, the highest price the bakery can charge, in dollars is $0.86623
