Question

The price of a diamond varies as the square of its weight. Three rings of equal weight, each composed of a diamond set in gold, have values pounds $a,b,c$, the diamond in them weighing $3,4,5$ carats respectively. Show that the value of a diamond of one carat is
pounds$(\frac{a+c}{2}-b)$,
the cost of workmanship being the same for each ring.

Answer 1

Let $V$ be the value of the diamond, $w$ be the weight of diamond, $e$ be the cost of workmanship

So,

$V=k{w}^2$

where $k$ is a constant.

Value for one carat is,

$V=k$

For Price(P) of ring,

$P=e+k{w}^2$

Then according to question,

$a=e+9k$

$b=e+16k$

$c=e+25k$

So,

$\frac{a+c}{2}-b=\frac{e+9k+e+25k}{2}-e-16k=\frac{2e+34k}{2}-e-16k$

$=e+17k-e-16k=k$ This is the price of one carat of diamond. Hence proved.