Finding Points of Intersection of a Parabolic Arch and a Hill

Answer: In summary, the conversation discusses the equation of a parabolic arch on a hill and finding its points of intersection. The equation of the arch is x^2 + 10y - 10 = 0 and the equation of the hill is y = 0.1x - 1. By substituting the hill equation into the arch equation, the points of intersection are found to be (4, -0.6) and (-5, -1.5). The speaker also mentions using the quadratic equation to find these points and expresses confusion about how to draw the arch. However, they later state that they have figured it out.
  • #1
physicsgal
164
0
"a parabolic arch has an equation x^2 + 10y - 10 = 0. the arch is on a hill with equation y = 0.1x-1. (measurements are in metres).

"find the points of intersection"
for this i substituted y =0.1x - 1 into the equation x^2 + 10y - 10 = 0:
x^2 + 10(0.1x - 1) - 10 = 0
x^2 + x - 10 - 10 = 0
x^2 + x - 20 = 0
should this -20 be 0 instead? (i dunno)


and then used the quadratic equation so x = 4 or x = -5.
(putting those amounts into y =0.1x - 1), i have intersections of (4, -0.6) and (-5, -1.5). are these correct?

also, i don't understand how to draw this. :yuck:

any help is appreciated!

~Amy
 
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  • #2
k, nevermind (figured it out) :)

~Amy
 
  • #3


I would say that the points of intersection calculated using the quadratic equation are correct. However, it is important to note that these points represent the x-coordinate of the intersections. To find the y-coordinate, you will need to plug in the x-values into the equation y = 0.1x - 1.

As for drawing the parabolic arch and hill, you can plot the points of intersection and then sketch the curves of the equations. Alternatively, you can use a graphing calculator or software to plot the equations and see the intersection points visually. It may also help to label the axes and include a scale to accurately represent the measurements in meters.

I hope this helps! Let me know if you have any further questions.
 

1. How do you determine the points of intersection between a parabolic arch and a hill?

To determine the points of intersection between a parabolic arch and a hill, you will need to use mathematical equations to represent the curve of the arch and the slope of the hill. Then, you can set these equations equal to each other and solve for the points of intersection. This can be done by hand or using a graphing calculator.

2. Are there any real-world applications for finding points of intersection between a parabolic arch and a hill?

Yes, there are many real-world applications for finding points of intersection between a parabolic arch and a hill. For example, architects and engineers may use this calculation to design bridges or tunnels that need to cross over a hill. It can also be used in landscaping to determine the best placement for retaining walls or walkways.

3. What factors can affect the accuracy of finding points of intersection between a parabolic arch and a hill?

The accuracy of finding points of intersection between a parabolic arch and a hill can be affected by various factors such as the precision of the mathematical equations used, the accuracy of the measurements taken for the arch and hill, and any external factors that may impact the shape of the hill or the arch.

4. Can points of intersection between a parabolic arch and a hill be found using any other methods?

Yes, points of intersection between a parabolic arch and a hill can also be found using numerical methods such as the bisection method or the Newton-Raphson method. These methods involve using a series of approximations to find the points of intersection.

5. Is there a specific formula or equation to find points of intersection between a parabolic arch and a hill?

No, there is no single formula or equation that can be used to find points of intersection between a parabolic arch and a hill. The method used will depend on the specific shape and characteristics of the arch and hill. However, mathematical equations and numerical methods can be used to accurately find these points of intersection.

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