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Math  /  Algebra

QuestionThe Kat Kit Chocolate company is doing market research to sell its Kat Kit Chocolate at the right price in order to increase profit. The profit equation for this chocolate bar is given by: y=−4x2+12x−5y=-4 x^{2}+12 x-5 y = Profit (Million of Dollars) x=x= Selling Price (Dollars) a) What should the company sell these chocolates in order to breakeven? (Zero profit) b) It is determined that the company needs to earn a profit of $3\$ 3 million to keep its investors happy. At what price(s) should it sell its Chocolate bars for? c) What price should the company sell its chocolates to maximize profits?

Studdy Solution

STEP 1

1. The profit equation is y=−4x2+12x−5 y = -4x^2 + 12x - 5 .
2. y y represents the profit in millions of dollars.
3. x x represents the selling price in dollars.
4. We need to find the selling price for breakeven, a specific profit, and maximum profit.

STEP 2

1. Solve for breakeven points (zero profit).
2. Solve for a specific profit of $3 million.
3. Determine the price for maximum profit.

STEP 3

To find the breakeven points, set the profit equation to zero: −4x2+12x−5=0 -4x^2 + 12x - 5 = 0

STEP 4

Use the quadratic formula to solve for x x : x=−b±b2−4ac2a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} where a=−4 a = -4 , b=12 b = 12 , and c=−5 c = -5 .

STEP 5

Calculate the discriminant: b2−4ac=122−4(−4)(−5)=144−80=64 b^2 - 4ac = 12^2 - 4(-4)(-5) = 144 - 80 = 64

STEP 6

Substitute the values into the quadratic formula: x=−12±64−8 x = \frac{-12 \pm \sqrt{64}}{-8}

STEP 7

Calculate the solutions: x=−12±8−8 x = \frac{-12 \pm 8}{-8} x1=−12+8−8=−4−8=12 x_1 = \frac{-12 + 8}{-8} = \frac{-4}{-8} = \frac{1}{2} x2=−12−8−8=−20−8=52 x_2 = \frac{-12 - 8}{-8} = \frac{-20}{-8} = \frac{5}{2}

STEP 8

To find the selling price for a profit of $3 million, set the profit equation to 3: \[ -4x^2 + 12x - 5 = 3 \]

STEP 9

Rearrange the equation: −4x2+12x−8=0 -4x^2 + 12x - 8 = 0

STEP 10

Use the quadratic formula again with a=−4 a = -4 , b=12 b = 12 , and c=−8 c = -8 .

STEP 11

Calculate the discriminant: b2−4ac=122−4(−4)(−8)=144−128=16 b^2 - 4ac = 12^2 - 4(-4)(-8) = 144 - 128 = 16

STEP 12

Substitute into the quadratic formula: x=−12±16−8 x = \frac{-12 \pm \sqrt{16}}{-8}

STEP 13

Calculate the solutions: x=−12±4−8 x = \frac{-12 \pm 4}{-8} x1=−12+4−8=−8−8=1 x_1 = \frac{-12 + 4}{-8} = \frac{-8}{-8} = 1 x2=−12−4−8=−16−8=2 x_2 = \frac{-12 - 4}{-8} = \frac{-16}{-8} = 2

STEP 14

To find the price that maximizes profit, find the vertex of the parabola, since it opens downwards.

STEP 15

The vertex x x -coordinate is given by: x=−b2a=−122(−4)=128=32 x = -\frac{b}{2a} = -\frac{12}{2(-4)} = \frac{12}{8} = \frac{3}{2}
The solutions are: a) Breakeven prices: x=12 x = \frac{1}{2} and x=52 x = \frac{5}{2} b) Prices for $3 million profit: \( x = 1 \) and \( x = 2 \) c) Price to maximize profit: \( x = \frac{3}{2} \)

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