Derivative Geometry Help (1 Viewer)

alussovsky · Nov 10, 2018

The question:

7. The sum of the radii of two circles is constant, so that r1 + r2 = k, where k is constant
(a) Find an expression for the sum of the areas of the circles in terms of one variable only
(b) Hence show that the sum of the areas is least when the circles are congruent

Thank you for your help!

fan96 · Nov 10, 2018

$A = \pi ({r_1}^2 + {r_2}^2)$

and

$k = r_1 + r_2 \implies r_2 = k - r_1$

noting that $k$ is a constant, we can then eliminate $r_2$ by doing one substitution.

alussovsky · Nov 10, 2018

fan96 said:
$A = \pi ({r_1}^2 + {r_2}^2)$

and

$k = r_1 + r_2 \implies r_2 = k - r_1$

noting that $k$ is a constant, we can then eliminate $r_2$ by doing one substitution.

Oh, I get that part now! Aaaahh, sorry, I think I was a bit out of it before, lol

But uhh, how would you do part b?

fan96 · Nov 10, 2018

alussovsky said:
Oh, I get that part now! Aaaahh, sorry, I think I was a bit out of it before, lol

But uhh, how would you do part b?

Part b) is a standard maximisation question.

$A$ will be a quadratic function of $r_1$ .

By inspecting the coefficient of ${r_1}^2$ we can see this is a concave-up ("smiley face") parabola and hence $A$ has a global minimum, though we will need to determine if this global minimum lies in the domain $r_1 > 0$ .

Because the maximum lies on a turning point, we may consider the derivative and solve

$\frac{dA}{d(r_1)} = 0$

You will eventually arrive at $r_1 = k/2$ .

HeroWise · Nov 11, 2018

Derivative Geometry Help (1 Viewer)

alussovsky

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fan96

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alussovsky

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fan96

617 pages

HeroWise

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