A paper drinking cup is being
designed in the shape shown in the accompanying figure.
The amount of paper needed to manufacture the cup is
determined by the surface area S of the cup, which is
given by s=πr√(r^(2))+h^(2)
where r is the radius and h is the height.

(b) Could the formula for S be simplified as follows?

πr√(r^(2))+h^(2)=πr(√(r^(2))+√(h^(2)))=πr(r+h)

Question

A paper drinking cup is being
designed in the shape shown in the accompanying figure.
The amount of paper needed to manufacture the cup is
determined by the surface area S of the cup, which is
given by s=πr√(r^(2))+h^(2)
where r is the radius and h is the height.

(b) Could the formula for S be simplified as follows?

πr√(r^(2))+h^(2)=πr(√(r^(2))+√(h^(2)))=πr(r+h)

anonymous · Answer

Nonono. In general
$$\sqrt{a + b}\ \not= \sqrt{a} + \sqrt{b} $$

anonymous · Answer

Could you explain your answer to me Mr. Newton?

anonymous · Answer

That simplification uses the assumption I listed above (splitting it up into two roots)

The assumption is false.

anonymous · Answer

Hmmm

anonymous · Answer

It probably got confused with
$$\sqrt{a\times b} = \sqrt{a} \times \sqrt{b} $$ which is true.

anonymous · Answer

So in the case of the paper, cup it false because it was split up into two roots?

anonymous · Answer

Yes.

In case you still don't believe me ¬_¬ try putting in some numbers for r and h.

sqrt(5^2 + 6^2) = sqrt(61) =/= 5 + 6

anonymous · Answer

Oh, I believe you.

anonymous · Answer

πr√(r^(2))+h^(2)=πr(√(r^(2))+√(h^(2)))=πr(r+h)