How Do You Solve This Word Problem. An Architect's Design For A Building Includes Some Large Pillars With Cross Sections In The Shape Of Hyperbolas. The Curves Can Be Modeled Be The Equation ((x^2)/.25)-((y^2/9))=1, Where The Units Are In Meters.?

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How Do You Solve This Word Problem. An Architect's Design For A Building Includes Some Large Pillars With Cross Sections In The Shape Of Hyperbolas. The Curves Can Be Modeled Be The Equation ((x^2)/.25)-((y^2/9))=1, Where The Units Are In Meters.?

2 Answers

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Answer 1

you are trying to relate to a 3 dimensional object with a 2 dimensional equation, not impossible but a royal pain in the kazoo. I have just spent over an hour on your question and have determined that I dont have enough information to work with. To the best of my calculations the width of the pillar at the top in not 120 cm but actually 147.19114 cm. How I arrived at these numbers is quite simple.
First solve for y on your graphing calculator, : Y = + or - (4 * x^2 - 1)^(9/2)
graph the two functions in a -2.5 to 2.5 by -2.5 to 2.5 window and trace.
This step you do the hard way because you trace x not y.
Therefore y = o when x = .5 remember that is only half the diameter, double it and you get 1meter or 100cm
so when we find y = 2 half the height x = .7359557 doubled to 1.4719114 meters. Not 1.2
Is there a faster way? Probably but I havent started calculus yet.
A thought occurs, plug in your y value on the home menu and solve for x and double.
Remembering that the equation puts the center of the pillar at the origin of a Cartesian Graph plane. ( hence the top of the 4 meter pillar is at y = 2)

Now that I have thoroughly confused and intimidated you, did I shed some light.

Answer 2

As Andre21198 says, the center of the pillar is located at the origin of the coordinate system. As you say, the narrowest point is at the vertical center, where y=0. The half-width of the pillar is presumed to be the x-value, so the width is twice the x value.

At the center, y=0, so we have
(x^2)/.25 - (0^2)/9 = 1
x^2 = .25 (multiply both sides by .25)
x = √.25 = .5 (take the square root of both sides)
width at the center = 2*.5 meters = 1.00 meters = 100 cm

At the top of the pillar, y=2, so we have
(x^2)/.25 - (2^2)/9 = 1
x^2 = .25(1 + 4/9) (add 4/9 to both sides, then multiply by .25)
x = √(13/36) = (√13)/6 (take the square root of both sides)
width at the top = 2*(√13)/6 meters ≈ 1.202 meters = 120.2 cm

(Because the equation is symmetrical about x=0 and about y=0, for every point on the curve (x, y), there are 3 other points also on the curve, (x, -y), (-x, y), and (-x, -y). The distance between the point (x, y) and the point (-x, y)--the width of the pillar at y--is 2x.)