Arch involving CONICS hyperbola equation

• aisha
In summary, you are to design a curved arch that is similar to a tunnel for cars. With a horizontal span of 100 meters and a maximum height of 20 meters, find the equation of a hyperbola that could represent the arch. After finding the equation, solve for k to get a and then your standard form equation for the hyperbola representing the arch will be complete.
aisha
Your task is to design a curved arch similar to the a tunnel for cars. with a horizontal span of 100 m and a maximum height of 20 m.

Using a domain of {x:-50<=x<=50} and {y:0<=y<=20} determine the following types of equations that could be used to model the curved arch.

the equation of a hyperbola in the form $$\frac {(x-h)^2} {a^2} - \frac {(y-k)^2} {b^2} = -1$$ where b=10 and the lower arm of the hyperbola would represent the arch.

How do i find a and the center (h,k) please help me I am struggling on this question.

Last edited:
i got h=0 and y=20 because the coordinate (0,20) is on the y-axis

a cancels out because 0/a^2 =0 so only left with k as unknown get a quadratic function and get two values for k, which value do I take? choose a point let's say (0,50) and find a, these are the steps i have so far.

Um for k -(20-k)^2 how do u expand this (-20+k) (-20+k) or (-20+k) (20-k)?

I didn't follow your problem,but i can tell u for sure that

$$-(20-k)^{2}=-(20-k)(20-k)=(20-k)(k-20)$$

Daniel.

ok I get $$\frac {k^2 +40 k-400} {100} =-1$$ is it possible to solve for k? once I find k i can plug it into the original equation to get a and then my standard form equation for the hyperbola representing the arch will be complete.

Sure you can solve for k. It's a simple quadratic. Multiply both sides by 100 and then use the quadratic formula.

hold on i changed what I did

$$-\frac {(20-k)^2} {100} = -1$$

ok I cross multiplied and got

$$400-40k+k^2=-100$$
$$k^2 -40k+500=0$$

this is my quadratic using the quadratic formula i keep getting a negative under the square root why? $$b^2-4(a)(c)$$ sqrt(-400)
teacher said that k=10 and 30 but I don't know how.

U can't get a negative under the square root.U should get 400.

Daniel.

P.S.It's +100 in the RHS.

it can't be +100 on the rhs because this is the equation of a hyperbola. Using the information givin I am trying to find the equation of the hyperbola that could represent a arch with a span of 100 metres and maximum height of 20 metres. I was trying to solve k so that I could sub this value into the original equation and then get a then my equation in standard form for the hyperbola will be complete.

Honey,from

$$-\frac{(20-k)^{2}}{100}=-1$$

please trust me that it follows

$$\frac{(20-k)^{2}}{100}=1 \Rightarrow (20-k)^{2}=100 \Rightarrow k_{1}=10,k_{2}=30$$

,okay,sweetheart?

Daniel.

the 100 is positive because you did what?

are we cross multiplying? or multiplying both sides by -100? what are we doing?

That equation i wrote has a negative sign too did u see that? before the brackets.

I simplified an equality through "-1".Or,if u prefer,i multiplied both sides through the same skinny "-1".

Daniel.

dextercioby said:
Honey,from

$$-\frac{(20-k)^{2}}{100}=-1$$

please trust me that it follows

$$\frac{(20-k)^{2}}{100}=1 \Rightarrow (20-k)^{2}=100 \Rightarrow k_{1}=10,k_{2}=30$$

,okay,sweetheart?

Daniel.

1. What is the equation for an arch involving a conic hyperbola?

The equation for an arch involving a conic hyperbola is y = a(cosh(x/a) - 1), where a is the distance from the center of the hyperbola to either focus and cosh is the hyperbolic cosine function.

2. How is the shape of an arch involving a conic hyperbola determined?

The shape of an arch involving a conic hyperbola is determined by the values of a and c, where a is the distance from the center to the focus and c is the distance from the center to the directrix. The larger the value of a, the narrower and taller the arch will be.

3. Can an arch involving a conic hyperbola have a negative a value?

Yes, an arch involving a conic hyperbola can have a negative a value. This will result in a reflection of the arch across the y-axis, with the opening facing to the left instead of the right.

4. How are arches involving conic hyperbolas used in real life?

Arches involving conic hyperbolas are commonly used in architecture, such as in the design of bridges, buildings, and other structures. They can also be seen in the shape of certain natural formations like rock arches and ocean waves.

5. Can the equation for an arch involving a conic hyperbola be simplified?

Yes, the equation for an arch involving a conic hyperbola can be simplified by factoring out the constant a. This will result in the equation y = a(cosh(x/a) - 1) becoming y = a(cosh(x/a) - a/a), which simplifies to y = acosh(x/a) - a.

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