# determine equation for parabola of best fit 4 data hourly rate 10 25 45 65 90 week earnings 700 1450 1890 1690 540 find vertex what makes most moneywhat hourly rate will make the most money round...

determine equation for parabola of best fit 4 data hourly rate 10 25 45 65 90 week earnings 700 1450 1890 1690 540 find vertex what makes most money

what hourly rate will make the most money round answer to nearest penny define vertex point

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### 1 Answer

Standard equation of a parabola is `y = ax^2+bx+c`

So we can use the given values to find the equation of the parabola.

700 = `a*10^2+b*10+c` = 100a+10b+c

700 = 100a+10b+c----------(1)

Similarly we can write other equations as ;

1450 = 625a+25b+c------(2)

1890 = 2025a+45b+c-------(3)

1690 = 4225a+65b+c----(4)

540 = 8100a+90b+c-----(5)

Solving (1),(2),(3) will give a=-0.8 , b= 78, c=0

Solving (2),(3),(4) will give a=-0.8 , b= 78, c=0

Solving (3),(4),(5) will give a=-0.8 , b= 78, c=0

So the all the points are best fitted to the parabola.

So equation of the parabola is y = -0.8x^2+78x

y = `-0.8x^2+78x`

= `-0.8[x^2-97.5x]`

= `-0.8[x^2-97.5x+(97.5/2)^2-(97.5/2)^2]`

= `-0.8[(x-97.5/2)^2-(97.5/2)^2]`

= `0.8xx(97.5/2)^2-0.8*(x-97.5/2)^2`

`(x-97.5/2)^2gt=0` always. So y will give its maximum when `(x-97.5/2)^2` is minimum.

Since `(x-97.5/2)^2gt=0` minimum is `(x-97.5/2)^2 = 0`

Then x = 97.5/2 = 48.75

**Maximum money will generate at a hourly rate of 48.75**

**The max amount of money is 0.8*(97.5/2)^2 = 1901.25**

**Sources:**