10 Raj is investigating how the price, \(P\) pounds, of a brilliant-cut diamond ring is related to the weight, \(C\) carats, of the diamond.
He believes that they are connected by a formula
$$P = a C ^ { n }$$
where \(a\) and \(n\) are constants.
10
- Express \(\ln P\) in terms of \(\ln C\).
10 - Raj researches the price of three brilliant-cut diamond rings on a website with the following results.
| \(\boldsymbol { C }\) | 0.60 | 1.15 | 1.50 |
| \(\boldsymbol { P }\) | 495 | 1200 | 1720 |
- Plot \(\ln P\) against \(\ln C\) for the three rings on the grid below.
\includegraphics[max width=\textwidth, alt={}, center]{091aecd0-d812-4a8f-8596-a1c91f3bae1c-15_1018_1467_751_283}
10
- (ii) Explain which feature of the plot suggests that Raj's belief may be correct.
10 - (iii) Using the graph on page 15 , estimate the value of \(a\) and the value of \(n\).
10
- Explain the significance of \(a\) in this context.
10 - Raj wants to buy a ring with a brilliant-cut diamond of weight 2 carats.
Estimate the price of such a ring.