The length of an isosceles triangle is increasing at a rate of 5 m/s. At what rate is the hypotenuse changing?

Question

anonymous · Answer

h= sqrt(2a^2)

anonymous · Answer

I can implicitly drive the above equation but I don't think that is what they want.

anonymous · Answer

(1/2)(2a^2)^(-1/2) * 4a = dh/da

anonymous · Answer

What equation relates the length of the triangle to its hypotenuse?

anonymous · Answer

like...?

anonymous · Answer

(sqrt(2a^2))^2/2 = area of triangle

ranga · Answer

|dw:1381017578039:dw|

anonymous · Answer

I was thinking that it was a right triangle.

ranga · Answer

I think the word "hypotenuse" implies a right triangle which also happens to be isosceles in this problem.

anonymous · Answer

ya it and isosceles

anonymous · Answer

an*

ranga · Answer

@znimon: Can you look at the diagram above and figure out what h is?

anonymous · Answer

sqrt2

ranga · Answer

yes $$\sqrt{2}$$
So if the side is of length s, then $$h = \sqrt{2}s$$
If s changes at the rate of 5m/s, then what is the rate of change of h?

anonymous · Answer

sqrt(2)*5

ranga · Answer

Yes.  $$5\sqrt{2}$$

anonymous · Answer

So you get to assume it is a right triangel?

anonymous · Answer

yes

wolf1728 · Answer

I figured I'd make a graphic that's oriented differently than the previous one.

anonymous · Answer

I got that the rate of change with respect to time for the hypotenuse is 5.

ranga · Answer

@wio: Is the word hypotenuse used in a context other than a right triangle?

wolf1728 · Answer

The again what is the length of an isosceles triangle - the height? the length of one side?

ranga · Answer

The length in this context refers to the two equal sides of the triangle with the third side being the hypotenuse.  This is a special case of a right triangle with two equal sides and hence the words hypotenuse and isosceles.

wolf1728 · Answer

Then it is definitely a 45 45 90 triangle right?

ranga · Answer

Yes.

wolf1728 · Answer

It seems to me (according to my graphic) that as the length (or height) of the triangle increases, the hypotenuse increases by twice that amount.

ranga · Answer

If you assume the words "length of an isosceles triangle" refers to the height of the triangle then you are right. 

But my interpretation is the length refers to the two equal sides of the isosceles triangle and the hypotenuse refers to the third side of the triangle. 

In the word problems I have seen, I have come across the phrase "the lengths of the sides of a triangle are a, b & c". But I have not come across the height of a triangle being referred to as the length of a triangle so far.