Question

mario has a business selling muffins. let x be the price of a muffin. then, the profit p for mario’s business is given by p(x) = -100x² + 350x - 150. choose the inequality that shows the business will make a positive profit. options: 0 ≥ -100x² + 350x - 150; 0 ≤ -100x² + 350x - 150; 0 > -100x² + 350x - 150; 0 < -100x² + 350x - 150 (with a checkmark). look at the factorization shown below. 0 < -100x² + 350x - 150; 0 < -50(2x² - 7x + 3); 0 < -50(2x - 1)(x - 3). select the range that mario can choose from to price his muffins and make a positive profit. options: x < $0.50; $0.50 < x < $3.00; $0.50 < x or x > $3.00; x > $3.00.

Explanation:

Step1: Analyze the profit function

The profit function is \( P(x) = -100x^2 + 350x - 150 \). For positive profit, \( P(x) > 0 \), so the inequality is \( 0 < -100x^2 + 350x - 150 \) (this part was already chosen correctly in the first sub - question).

Step2: Analyze the factored form

We have the factored form \( 0 < - 50(2x - 1)(x - 3) \). First, divide both sides by - 50. Remember that when dividing an inequality by a negative number, the direction of the inequality sign changes. So we get \( 0> (2x - 1)(x - 3) \) (because \( \frac{0}{-50}=0 \) and dividing by - 50 flips the inequality).

Step3: Find the roots of the quadratic factor

Set \( (2x - 1)(x - 3)=0 \). Then \( 2x - 1 = 0\) gives \( x=\frac{1}{2}=0.5 \) and \( x - 3=0\) gives \( x = 3 \).

Step4: Analyze the sign of the quadratic

The quadratic function \( y=(2x - 1)(x - 3)=2x^2-7x + 3 \) is a parabola opening upwards (since the coefficient of \( x^2 \) is positive). We want to find where \( (2x - 1)(x - 3)<0 \) (because we had \( 0>(2x - 1)(x - 3) \)). For a parabola opening upwards, the function is negative between its two roots. So the solution to \( (2x - 1)(x - 3)<0 \) is \( 0.5

Answer:

For the first sub - question (choosing the inequality for positive profit): \( 0 < -100x^{2}+350x - 150 \)

For the second sub - question (choosing the price range): \( \$0.50 < x < \$3.00 \)