GED Mathematical Reasoning TestAlgebraic Problem Solving with Expressions and EquationsHard
A civil engineer is designing a parabolic arch for a pedestrian bridge. The height of the arch, h, in meters, above the ground at a horizontal distance, x, in meters, from the center of the arch is given by the equation h(x) = -0.1x^2 + 10. For what range of x-values is the height of the arch at least 5 meters?
- A-5 ≤ x ≤ 5
- B-10 ≤ x ≤ 10
- C-7.07 ≤ x ≤ 7.07
- Dx ≤ -7.07 or x ≥ 7.07
Show answer & explanationAnswer & explanation
Correct answer: C. -7.07 ≤ x ≤ 7.07
We need to find the range of x-values for which h(x) ≥ 5. So, -0.1x^2 + 10 ≥ 5. Subtract 10 from both sides: -0.1x^2 ≥ -5. Divide by -0.1 (remember to reverse the inequality sign when dividing by a negative number): x^2 ≤ 50. Take the square root of both sides: √x^2 ≤ √50. This means |x| ≤ √50. Since √50 is approximately 7.07, the inequality becomes |x| ≤ 7.07. This translates to -7.07 ≤ x ≤ 7.07.
Why the other options are wrong
- A. This is an incorrect range, possibly from x^2 <= 25 or other miscalculation.
- B. This would be the range if the height was 0 (the endpoints of the arch), or if the inequality was x^2 <= 100.
- D. This represents the range where the height is less than 5 meters, which is the opposite of what is asked.
Solving Quadratic Inequalities
Finding the range(s) of values for the variable that satisfy an inequality involving a quadratic expression, often by finding roots and testing intervals.
- Set inequality to zero.
- Find roots of the quadratic equation.
- Test intervals or use parabola's shape.
Memory trick: Roots, Test, Shade: Find where the parabola crosses, then check the sections!