Question

Assuming that the values of diamonds are proportional, other things being equal, to the squares of their weights, and that a certain diamond which weighs one carat is worth \(\$ m\), show that it is safe to pay at least \(\$ 8 m\) for two diamonds which together weigh 4 carats, if they are of the same quality as the one mentioned.

Short answer

Answer: Yes, it is safe to pay at least $8m for the two diamonds which together weigh 4 carats.

Step-by-step solution

Find the proportionality formula

Since the value of the diamonds (V) is proportional to the squares of their weights (W), we can write the proportionality formula as: $$ V=kW^2 $$ Where \(k\) is the proportionality constant.

Calculate the value of k

Given a diamond worth \(m\) dollars and weighs 1 carat, we can plug those values into the formula and solve for \(k\): $$ m=k(1)^2 $$ So the proportionality constant is \(k=m\) and the proportionality formula becomes: $$ V=mW^2 $$

Minimum values for the 4-carat diamonds

We need to find two diamonds weighing 4 carats in total when added together. Let's label the weights of the two diamonds as \(W_1\) and \(W_2\), and their values as \(V_1\) and \(V_2\). The minimum value for the diamonds would be when they have equal weights. Therefore, \(W_1=W_2=2 \,\mathrm{carats}\). Now we can calculate the values of the two diamonds. $$ V_1 = m(W_1)^2 = m(2)^2 = 4m $$ $$ V_2 = m(W_2)^2 = m(2)^2 = 4m $$

Calculate the total value of the 4-carat diamonds

Adding the values of \(V_1\) and \(V_2\), we get the total value of the two diamonds: $$ V_\mathrm{total} = V_1 + V_2 = 4m + 4m = 8m $$

Compare the total value with 8m dollars

We have found that the value of the two diamonds, when each weighs 2 carats and is of the same quality as the 1-carat diamond, is \(8m\). Since this is the minimum value, it is safe to pay at least \(\$8m\) for the two diamonds which together weigh 4 carats.

Key concepts

Value of Diamonds

The value of diamonds is not just about aesthetics and rarity, it’s crucially influenced by their weight. According to the problem, the value (\(V\)) is directly proportional to the square of the diamond's weight (\(W\)). This means that if a diamond becomes heavier, its value increases exponentially, not just linearly. The formula representing this relationship is \(V = kW^2\), where \(k\) is the proportionality constant. This concept helps in understanding why larger diamonds might fetch much higher prices, despite being just a little heavier. In practical terms, if a 1-carat diamond is worth \(m\) dollars, then a diamond that weighs twice as much (2 carats) doesn't just double in value; it becomes four times as valuable, due to the squaring of the weight. This relationship is key to solving the problem of finding when it is safe to purchase heavier diamonds based on their combined weight and value.

Weights of Diamonds

The weight of a diamond, measured in carats, is often viewed as an indicator of value. However, in this mathematical challenge, we look at weights in terms of their effect on the diamond’s value. Specifically, we explore how splitting total weight can influence the overall value. In our exercise, two diamonds together weigh 4 carats. By calculating their combined value using their square weights, we see an interesting phenomenon. If both diamonds weighed 2 carats each, their individual contributions to the total value become predictable using our formula \(V = mW^2\). This understanding allows us to confidently say that splitting 4 carats into two equal parts gives us a minimum assured value that is quite favorable. Thus, the weights of diamonds are not just about measuring physical mass but are central to predicting and calculating possible returns in their sale.

Proportionality Constant

The proportionality constant, \(k\), is a linchpin in translating the relationship between a diamond's weight and its value into a fixed equation. In our task, initially, a diamond weighing 1 carat has a worth of \(m\) dollars. By substituting these known values into the proportion formula \(V = kW^2\), we determine that \(k = m\). This discovery simplifies our calculations, turning the formula into \(V = mW^2\). Understanding this constant allows us to directly apply given data to predict diamond values for any weight. The benefit of pinpointing \(k\) is that it ensures consistency and accuracy in assessing value across varying diamond weights, abiding by the established proportional relationship. It is a straightforward yet powerful tool for anyone wishing to delve into the intricate world of diamond valuation.

Mathematical Problem Solving

Mathematical problem solving involves breaking down a problem into manageable parts, applying known mathematical principles, and carefully piecing together a solution. In our case, this involved using the formula \(V = mW^2\) to evaluate the value of clustered weights for multiple diamonds.First, we determined the foundational relationship between weight and value. Then, recognizing that the maximum value arises when the weights were divided evenly, gave us a logical path forward. By calculating each diamond's potential minimum value of 2 carats to be 4\(m\), and then summing these for the total weight of 4 carats, we arrived at a solution suggesting it was safe to pay up to 8\(m\) dollars for both diamonds.This problem-solving process emphasizes not just the algebraic manipulations but also the contextual understanding of why these steps are taken. It turns a formula into a practical decision-making tool. For students, embracing these strategies enhances both their analytical skills and their confidence in approaching complex mathematical challenges.