The quadratic function $y=-x^2+10 x-8$ models the height of a trestle on a bridge. The $x$-axis represents ground level.
To find where the section of the bridge meets ground level, solve $0=-x^2+10 x-8$.
Where does this section of the bridge meet ground level?
Choose an equation that would be used to solve $0=-x^2+10 x-8$.
$(x-5)^2=17$
$(x-10)^2=25$
$(x+25)^2=-8$
Solve the equation to find where the trestle meets ground level. Enter your answers from least to greatest and round to the nearest tenth.
The trestle meets the ground at $\square$ units and $\square$ units.
Answer (2)
Recognize the problem as solving a quadratic equation to find where the trestle meets the ground.
Solve the equation ( x − 5 ) 2 = 17 by taking the square root of both sides, resulting in x = 5 ± 17 .
Calculate the two values of x : x 1 = 5 − 17 ≈ 0.877 and x 2 = 5 + 17 ≈ 9.123 .
Round the solutions to the nearest tenth, giving the points where the trestle meets the ground: 0.9 and 9.1 .
Explanation
Understanding the Problem We are given the quadratic function y = − x 2 + 10 x − 8 which models the height of a trestle on a bridge. The x -axis represents ground level. We want to find where the trestle meets the ground, which means we need to solve the equation 0 = − x 2 + 10 x − 8 . We are given the equation ( x − 5 ) 2 = 17 which is equivalent to the original equation.
Solving the Equation We need to solve the equation ( x − 5 ) 2 = 17 . To do this, we take the square root of both sides of the equation: ( x − 5 ) 2 = ± 17 x − 5 = ± 17 Now, we solve for x :
x = 5 ± 17
Calculating the Roots We have two possible solutions for x :
x 1 = 5 − 17 x 2 = 5 + 17 We need to approximate these values to the nearest tenth. We know that 17 ≈ 4.123 . Therefore, x 1 ≈ 5 − 4.123 = 0.877 x 2 ≈ 5 + 4.123 = 9.123
Finding the Ground Intersections Rounding to the nearest tenth, we get: x 1 ≈ 0.9 x 2 ≈ 9.1 So, the trestle meets the ground at approximately 0.9 units and 9.1 units.
Final Answer The trestle meets the ground at approximately 0.9 units and 9.1 units.
Examples
Understanding where a bridge's trestle meets the ground is crucial for safety and stability. This calculation helps engineers determine the exact points of contact, ensuring the bridge is securely anchored. Knowing these points allows for precise construction and maintenance, preventing potential structural issues. This problem demonstrates how quadratic equations can be applied to real-world engineering challenges, ensuring the longevity and safety of infrastructure.
We solve the equation ( x − 5 ) 2 = 17 to find where the trestle meets ground level. The solutions for x are approximately 0.9 units and 9.1 units. Therefore, the trestle meets the ground at these two points.
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