Answer :
Answer:
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Given that the price of a Burmese sapphire is proportional to the cube of its weight, we can express the price \( P \) as:
\[ P = k \cdot W^3 \]
where \( k \) is a proportionality constant and \( W \) is the weight of the sapphire.
Let's denote the original weight of the sapphire by \( W \).
The price of the original sapphire is given by:
\[ P_{\text{original}} = k \cdot W^3 = 10000 \]
Now, the sapphire is broken into two pieces with a weight ratio of 2:3. Let the weights of these pieces be \( W_1 \) and \( W_2 \) respectively. Since their ratio is 2:3, we have:
\[ W_1 = \frac{2}{5} W \]
\[ W_2 = \frac{3}{5} W \]
We need to find the new prices of these two pieces after the sapphire is broken.
The price of the first piece is:
\[ P_1 = k \cdot W_1^3 = k \left( \frac{2}{5} W \right)^3 = k \cdot \frac{8}{125} W^3 \]
The price of the second piece is:
\[ P_2 = k \cdot W_2^3 = k \left( \frac{3}{5} W \right)^3 = k \cdot \frac{27}{125} W^3 \]
The combined price of both pieces is:
\[ P_{\text{new}} = P_1 + P_2 = k \cdot \frac{8}{125} W^3 + k \cdot \frac{27}{125} W^3 \]
\[ P_{\text{new}} = k \left( \frac{8}{125} W^3 + \frac{27}{125} W^3 \right) \]
\[ P_{\text{new}} = k \cdot \frac{35}{125} W^3 \]
\[ P_{\text{new}} = k \cdot \frac{7}{25} W^3 \]
We know that \( k \cdot W^3 = 10000 \), so:
\[ P_{\text{new}} = 10000 \cdot \frac{7}{25} \]
\[ P_{\text{new}} = 10000 \cdot 0.28 \]
\[ P_{\text{new}} = 2800 \]
Therefore, the loss Sophia incurred is:
\[ \text{Loss} = P_{\text{original}} - P_{\text{new}} \]
\[ \text{Loss} = 10000 - 2800 \]
\[ \text{Loss} = 7200 \]
Thus, Sophia lost $7200.
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Answer:
Sophia lost approximately $9,979.51
Step-by-step explanation:
To calculate how much Sophia lost when the Burmese sapphire broke, we can use the concept that the price of the sapphire is proportional to the cube of its weight. Since the weight ratio of the two pieces is 2:3, let's assume the weights of the two pieces are 2x and 3x, respectively.
Given that the original sapphire was worth $10,000, we can set up an equation based on the cube of the weights and the value:
Original sapphire value = $10,000
Weight ratio = 2:3
Let the weight of the smaller piece be 2x and the weight of the larger piece be 3x.
The total weight of the original sapphire = 2x + 3x = 5x
The price is proportional to the cube of the weight, so:
(2x)^3 + (3x)^3 = $10,000
Solving the equation:
8x^3 + 27x^3 = $10,000
35x^3 = $10,000
x^3 = $10,000 / 35
x^3 = $285.71
x ≈ $6.83
Now, we can find the value of the larger piece:
3x = 3 * $6.83 ≈ $20.49
The value Sophia lost is the difference between the original sapphire's value and the value of the larger piece:
$10,000 - $20.49 ≈ $9,979.51
Sophia lost approximately $9,979.51 when the Burmese sapphire broke into two pieces of different sizes.