# using differentials to estimate the increase in area

"A rectangle is inscribed in a semicircle of radius $5$m. Estimate the increase in area of the rectangle using differentials if the length of the base along the diameter is increased from $6$m to $6\frac{1}{6}$ m."

Now the problem I have is figuring out how to sort all this information. First of all, is the semi circle even relevant? It tells me that the base of the RECTANGLE is increasing, so I don't really think there is any need to consider the semi circle. I could be wrong about that however.

Can someone nudge me in the right direction for this?

I think what I have to do is figure out what the area of the rectangle in terms of the semi circle parameters but I am not 100% sure about that.

Visually, I'm guessing I have this sort of situation...

First of all, is the semicircle even relevant?

Yes, increasing the base length of the rectangle will decrease the height, since the top corners of the rectangle are constrained to lie on the semicircle.

I think what I have to do is figure out what the area of the rectangle in terms of the semi circle parameters.

That is correct. If you imagine the semicircle with its base along the $x$-axis, with the center of the base at the origin, then the equation of the top of the semicircle is $y = \sqrt{25 - x^2}$. An expression for the area of the rectangle is $A = 2xy$. Using these expressions, you can obtain an expression for the area of the rectangle in terms of $x$. This looks like $A(x) = 2x\sqrt{25 - x^2}$.

You are interested in rates of change. So now you want to differentiate. On the left you will have $\frac{dA}{dx}$ and on the right you'll have the derivative that you calculate. Finally, "multiply" by $dx$ on both sides to get the differential form. Then it is just a matter of plugging in $x$ and $dx$ from the problem statement to get $dA$.

• So I managed to figure it all out and obtain $dA = \frac{50-4x^2}{\sqrt{25-x^2}} dx$ . My $dx$ value is simply $\frac{1}{6}$. However, what is my $x$ value? It can't be $6$, because if I do that, my denominator is complex and areas are real so I am not sure if that is right... I am going to assume that $x=3$ instead since the parameters are in terms of the radius now. Is that right? – Future Math person Nov 5 '17 at 22:41
• Yes $x$ should be $3$ since it is only half of the base length of the triangle (remember the origin is the center of the base). And your $dA$ looks right to me as well. – wgrenard Nov 5 '17 at 22:48
• Thanks a lot! I think I got it now... – Future Math person Nov 5 '17 at 22:55