Question

The profit, in dollars, made by selling x bottles of 100% All-Natural Certified Free-Trade Organic Sasquatch Tonic is given byP(x)=−x^2+35x−120 for 0≤x≤45.

Answer 1

Real life situations can be modeled using various types of functions.

When 25 bottles are sold, the profit made would be $130

The profit function is given as:

$\mathbf{f(x) = -x^2 + 35x - 120}$

The complete question requires, the number of bottles that ensures a profit of  $130.

This means that:

$\mathbf{f(x) = 130}$

So, we have:

$\mathbf{130 = -x^2 + 35x - 120}$

Collect like terms

$\mathbf{0 = -x^2 + 35x - 120 - 130}$

$\mathbf{0 = -x^2 + 35x - 250}$

Expand

$\mathbf{0 = -x^2 + 25x + 10x - 250}$

Factorize

$\mathbf{0 = -x(x - 25) + 10(x - 25)}$

Factor out x - 25

$\mathbf{0 = (-x + 10)(x - 25)}$

Solve for x

$\mathbf{(-x + 10) = 0\ or\ (x - 25) = 0}$

$\mathbf{x = -10\ or\ x =25}$

The number of bottles cannot be negative.

So:

$\mathbf{x =25}$

Hence, 25 bottles would ensure a profit of $130

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