Praxis Core Academic Skills for Educators: Mathematics (5733)Algebra and FunctionsHard
A production line's daily output, Q, is related to the number of hours, h, it operates by the equation Q = 50h - h^2. For what range of hours operated will the daily output be at least 400 units?
- A10 ≤ h ≤ 40
- Bh ≤ 10 or h ≥ 40
- C0 ≤ h ≤ 10
- Dh ≥ 40
Show answer & explanationAnswer & explanation
Correct answer: A. 10 ≤ h ≤ 40
To find when the output is at least 400 units, we set up the inequality 50h - h^2 ≥ 400. Rearranging gives h^2 - 50h + 400 ≤ 0. Factoring the quadratic, we look for two numbers that multiply to 400 and add to -50. These are -10 and -40. So, (h - 10)(h - 40) ≤ 0. The roots are h = 10 and h = 40. For the quadratic to be less than or equal to zero, h must be between the roots, inclusive: 10 ≤ h ≤ 40.
Why the other options are wrong
- B. This represents the range where the output is less than 400, not at least 400.
- C. This range would yield output less than 400 (except at h=10) or not maximal.
- D. This range would yield output less than 400 (except at h=40) or negative.
Solving Quadratic Inequalities
Solving a quadratic inequality involves finding the range(s) of values for the variable that satisfy the inequality. This often includes finding the roots of the associated quadratic equation and testing intervals.
- Rearrange the inequality to have zero on one side.
- Find the roots of the corresponding quadratic equation.
- Use a sign chart or test points in intervals defined by the roots.
Memory trick: Find the Zeros, Test the Zones!