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amathproblemthatneedsolve

amathproblemthatneedsolve

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I found this question online but didn't see an answer so I want to try and solve it:

Information is given:

• The swimming pool is 18 m wide and is surrounded by a path.
• The roof needs to be at least 6 m above any point on the water in the pool.
• The roof must not be more than 8.5 m high.
• The maximum width of the roof for the swimming pool complex allowed is 36 m.

You must use the equation for the parabola in the form [MATH]y^2=4ax[/MATH]
Design A has a parabolic cross-section. Its maximum height is 8.5 m and the focus of the parabola is 1.5 m above the centre of the pool.

• Find a mathematical model for the design .
• Check whether or not the height of the roof is at least 6 m above the pool at any point.
• Check whether or not the width of the pool roof is less the 36 metres.

What I know:

d=18: width of the pool

D2=36: maximal width of the roof

h=6: minimal height of roof over the water

H2=8.5: maximal height of the roof

I'm not really sure how to progress I would assume I need find the focus of the parabola but unsure how to do this?
 
amathproblemthatneedsolve said:
I found this question online but didn't see an answer so I want to try and solve it:

Information is given:

• The swimming pool is 18 m wide and is surrounded by a path.
• The roof needs to be at least 6 m above any point on the water in the pool.
• The roof must not be more than 8.5 m high.
• The maximum width of the roof for the swimming pool complex allowed is 36 m.

You must use the equation for the parabola in the form [MATH]y^2=4ax[/MATH]
Design A has a parabolic cross-section. Its maximum height is 8.5 m and the focus of the parabola is 1.5 m above the centre of the pool.

• Find a mathematical model for the design .
• Check whether or not the height of the roof is at least 6 m above the pool at any point.
• Check whether or not the width of the pool roof is less the 36 metres.

What I know:

d=18: width of the pool

D2=36: maximal width of the roof

h=6: minimal height of roof over the water

H2=8.5: maximal height of the roof

I'm not really sure how to progress I would assume I need find the focus of the parabola but unsure how to do this?
Click to expand...
Start with a drawing.
 
amathproblemthatneedsolve said:
… You must use the equation for the parabola in the form [MATH]y^2=4ax[/MATH]
[The roof] has a parabolic cross-section …
Click to expand...
Hello amptns. Are you sure that is the given form? The graph of that form is a parabola that opens to the right. I would expect the ground to be horizontal.

?
 
Otis said:
Hello amptns. Are you sure that is the given form? The graph of that form is a parabola that opens to the right. I would expect the ground to be horizontal.

?
Click to expand...
Indeed that's the case......Weird. It specifically states not to use [MATH]y=Ax^2+B[/MATH]
 
amathproblemthatneedsolve said:
Sorry for the poor quality …
Click to expand...
I'm not sure what you mean about the quality, amptns, although I would have labeled the 8.5 and 1.5 distances differently.

The horizontal line representing the ground needs to be vertical. Rotate your diagram 90º counterclockwise. The ground-line is now x=8.5, and the peak of the roof (i.e., the vertex of the parabola) is at the Origin. We assume that the water surface is level with the ground.

x is the distance from the roof's peak toward the water's surface.

y is the perpendicular distance from the parabola's axis of symmetry to the parabola (the roof), at each x in the restricted domain.

EDIT: Ignore the rest of this post; it's not good information. See post #12.

Let's assume that the roof's peak is exactly 8.5 meters above the water's surface (as shown on your diagram). Let's assume also that the roof's width at the ground is exactly 36 meters. With those assumptions, determine the value of a. Does that value ensure the other requirement is met?

If it doesn't, then in what direction would we need to shift the line x=8.5?

?
 
After all my working I get the below. I’m pretty sure this is correct please advise if not, it seems to satisfies the requirements. ( is it in the right equation form?)
[MATH]y=8.5-0.03086x^2[/MATH]
 
amathproblemthatneedsolve said:
… It specifically states not to use [MATH]y=Ax^2+B[/MATH]
Click to expand...
amathproblemthatneedsolve said:
… is [this] in the right equation form? …
[MATH]y=8.5-0.03086x^2[/MATH]
Click to expand...
Hi amptns. It's not the correct form.

y = -0.03086x^2 + 8.5

That matches the form you were told not to use:

y = Ax^2 + B
A = -0.03086
B = 8.5

It's not necessary to express y in terms of x. You may work with the given form, as is.

However, if you would like to work with half the parabola, then you can get y as a function of x. In that case, start by solving the given form for y.

?
 
Hey amptns. Despite seeing your diagram, I'd forgotten that the focus was set in the given information, so it was incorrect of me to post earlier that we could find the value of a by assuming the roof width was exactly 36 meters. I'd also read more into the exercise than what you posted. We're not asked to fix a faulty design, so it was unnecessary for me to post earlier about moving the line x=8.5, as well. (Hmm, the phone just started ringing; it's probably Jomo.)

Okay, let's back up to the given form:

y^2 = 4ax

From the focus information, you'd found that a=7, yes? Therefore, all that's needed to "find a mathematical model for the design" is to substitute a=7 in the given form.

Next, use that model to check whether the roof height is at least 6 meters above the pool at any point and also to check whether the roof width is not more than 36 meters.

If either of those requirements is not satisfied by that model, then we simply say so and we're done.

?
 
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