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Inequality Question

Hi, I know that this is homework, but I would be indebted to whoever can point out and explain where I have gone wrong and how I should have gone right:

The area of a rectangle is 12 cm². Find the range of possible values of the width of the rectangle if the diagonal is more than 5cm.

I got (apologies for my inability to use the Latex notation).

height=h, width=w

h^2 + w^2 > 5^2

h^2 + w^2 > 25

hw = 12

h = 12/w

(12/w)^2 + w^2 > 25

144/w^2 + w^2 > 25

144 + w^4 > 25w^2

Then the whole thing falls apart for me.

Thank you for your help, —Daniel (‽) 09:46, 30 September 2006 (UTC)

Define z = w^2. Then your last equation can be rewritten as 144 + z^2 > 25z, or equivalently z^2 − 25z + 144 > 0. This is a standard quadratic inequation. Given the range of z satisfying it, and considering that w must be non-negative, the possible range for w consists of the square roots of the non-negative part of the solution range for z. It may be easier to look at the values that violate the inequation.

A different route to the solution is found from the consideration that you can easily deduce inequations for h^2 + 2hw + w^2 = (h + w)^2 as well as for h^2 − 2hw + w^2 = (h − w)^2.  --Lambiam^Talk 10:21, 30 September 2006 (UTC)

Thanks a lot. I used your first method to come to 0 < w < 3 (or) w > 3. I don't quite understand what you mean by the second method. Thank you again! —Daniel (‽) 16:06, 30 September 2006 (UTC)

Unfortunately, I think you made an error somewhere. Take w = 2√3 = 3.4541... This satisfies w > 3. Then w^2 = 12, so h = 12/w = w. Then h^2 + w^2 = 2w^2 = 24, which is not greater than 25. Unless you made a simple copying error, to get into this situation, you must have concluded before that z > 9 satisfies the inequation in z. But clearly that is incorrect: 144 + 100 < 250.  --Lambiam^Talk 16:25, 30 September 2006 (UTC)

I suspect the problem is meant to suggest a 3-4-5 right triangle, which is half a rectangle with diagonal 5 and area 12. Suppose we let the width be 4, the larger side. If the width increases by a factor of s, the height must be divided by s to maintain the area. Consider the effect on the diagonal. Is an width increase allowed? A decrease? (Hint: if the rectangle is a perfect square, the sides have length √12, making the diagonal too short; while if the width is at least 5 the diagonal is at least 5.) --KSmrq^T 18:34, 30 September 2006 (UTC)

Lambiam: Sorry, the keyboard gremlin got me there; I meant to type 0 < w < 3 (or) w > 4. Hopefully that sorts out your problem. KSmrq: I can vaguelly see where you're going, but to be honest lack the time and energy to follow it up, as I have answered the question and have a huge amount of other stuff to do. :( —Daniel (‽) 20:03, 1 October 2006 (UTC)

Venn diagrams

I was wondering, if all wigs are wags, and some wags are wogs, does that mean that all wigs are wogs? Please hurry, my exam finishes in 10 minutes!

If all students are humans, and some humans are females, does that mean that all students are females?  --Lambiam^Talk 15:51, 30 September 2006 (UTC)

If your exam finishes in 10 minutes, what are you doing writing on Wikipedia? —Daniel (‽) 16:02, 30 September 2006 (UTC)

No. Himanyo 17:27, 30 September 2006 (UTC)

They were doing the Tickle IQ test, and just failed miserably. Don't worry, if you got over 100, you are average. But seriously dude, who tries to cheat on an IQ test? --AstoVidatu 00:04, 1 October 2006 (UTC)

Isn't that IQ test which is so ridiculously unbalanced that the average score is about 120? Laïka 09:42, 1 October 2006 (UTC)

If all students are slackers, and some slackers don't do their own homework, does that mean no students do their own homework ? StuRat 11:34, 1 October 2006 (UTC)

Gold!! Thanks for your help guys, that was a past question from a math test I did. PS. I got it right. —The preceding unsigned comment was added by 83.142.184.86 (talk • contribs) 14:48, October 1, 2006 UTC).

For further in-depth understanding, you may try Syllogism. --Lemontea 14:12, 3 October 2006 (UTC)

Function question

I was wondering if I could find an equation to describe a curve in a Cartesian System of Axes, given that the curve I need is the graphs of 2 (or maybe more) functions combined.

i.e. for:

f(x) = x^2
g(x) = sqrt(x)

I would need an equation to describe a curve like the one at the right

Thanks, --Danielsavoiu 15:50, 30 September 2006 (UTC)

You could use the equation (x − y^2)(y − x^2) = 0, which however does not give you a curve. If you want to stay within the first quadrant, gluing these two graphs together, you could use a parametric equation, something like

     x = t^2 if t < 0, x = t  otherwise

     y = − t if t < 0, y = t^2 otherwise

 --Lambiam^Talk 15:59, 30 September 2006 (UTC)

Thanks, Lambiam. I thought about parametric equations, but I would rather have only one solid equation. Your first suggestion ( (x − y^2)(y − x^2) = 0 ) is ok, but what if I wanted to glue the graphs of:

f(x) = x^3
g(x) = 3x

--Danielsavoiu 16:17, 30 September 2006 (UTC)

If I understood your question correctly, then no you can't do that. There's no single function of one variable that can have the values of x^3 and 3x at the same time. You can, however, glue then in different sides of the Y axis: $x^{3}\left({\frac {\operatorname {sgn}(x)+1}{2}}\right)+3x\left({\frac {\operatorname {sgn}(-x)+1}{2}}\right)$.

Where sgn() is the sign function. This works with any function. The functions are "glued", but their domains aren't overlapping like you seem to want. ☢ Ҡi∊ff⌇↯ 17:08, 30 September 2006 (UTC)

If you want to combine the graphs of y = x^3 and y = 3x, use (y − x^3)(y − 3x) = 0.  --Lambiam^Talk 23:28, 30 September 2006 (UTC)