$\begin{array}{l} V=\frac{2}{3}+64 \\ V=\frac{2}{3}+\frac{162}{3} \\ V=\frac{164}{3} m^3 \end{array}$ What is Amie's error? A. Amie should have multiplied 54 by $\frac{2}{3}$. B. Amie should have multiplied 54 by $\frac{4}{3} \pi$. C. Amie should have multiplied 54 by $\frac{4}{3}$.
Answer (2)
Amie incorrectly calculated V as 3 2 + 64 = 3 164 m 3 .
The options suggest Amie should have multiplied 54 by some factor.
The most plausible error is related to the volume of a sphere, V = 3 4 π r 3 .
Therefore, Amie should have multiplied 54 by 3 4 π .
Explanation
Analyzing the Calculation Let's analyze Amie's calculation to identify the error. The calculation is presented as follows:
V = 3 2 + 64 V = 3 2 + 3 162 V = 3 164 m 3
We need to determine what Amie was trying to calculate and where she made a mistake. From the second line, we see that 64 was converted to a fraction with a denominator of 3. Let's check if that conversion is correct: 64 = 3 64 × 3 = 3 192 . However, Amie wrote 3 162 . This indicates an error in converting 64 to a fraction with a denominator of 3. The options suggest that Amie should have multiplied 54 by some factor. Let's examine the relationship between 64 and 54. It seems there's no direct relation, so let's focus on the conversion error.
Identifying the Error The error is in the conversion of 64 to a fraction with a denominator of 3. Amie wrote 3 162 instead of 3 192 . This means she incorrectly calculated 64 × 3 as 162 instead of 192. The options suggest that Amie should have multiplied 54 by some factor. Let's consider the possibility that the problem involves a sphere's volume, which has a factor of 3 4 π or 3 4 . If the problem involved a sphere with a radius related to 54, then the options might make sense. However, based on the given calculation, it's more likely that the error is simply in the arithmetic.
Evaluating the Options Since Amie wrote 3 162 instead of 3 192 , and 162/3 = 54 , it is possible that the original equation was V = 3 2 + 54 . However, the problem states that V = 3 2 + 64 , so the error is still in the conversion. Let's analyze the options:
Amie should have multiplied 54 by 3 2 . This doesn't seem directly related to the error.
Amie should have multiplied 54 by 3 4 π . This suggests a sphere volume calculation, but it doesn't explain the arithmetic error.
Amie should have multiplied 54 by 3 4 . This also suggests a sphere volume calculation, but it doesn't explain the arithmetic error.
Given the initial equation V = 3 2 + 64 , the most direct error is that Amie incorrectly converted 64 to 3 162 instead of 3 192 . However, none of the options directly address this. The closest option is that Amie should have multiplied 54 by some factor, implying that 54 is relevant to the problem. Since 64 × 3 = 192 and 54 × 3 = 162 , the error is that Amie used 54 instead of 64 in some context. However, without more information, it's difficult to pinpoint the exact error.
Determining the Most Plausible Error Based on the given information, it's most likely that the problem involves a sphere and Amie made an error in calculating the volume. The volume of a sphere is given by V = 3 4 π r 3 . If r 3 = 54 , then the volume would involve multiplying 54 by 3 4 π . Therefore, the most plausible error is that Amie should have multiplied 54 by 3 4 π .
Examples
Understanding errors in calculations is crucial in many real-life situations. For example, when calculating the amount of ingredients needed for a recipe, an error in multiplication or addition can lead to a dish that doesn't taste right. Similarly, in construction, incorrect measurements can result in structural problems. In finance, errors in calculating interest or loan payments can have significant financial consequences. Therefore, being able to identify and correct errors is a valuable skill in various fields.
Amie's calculation error lies in miscalculating the addition of 64 as 3 162 instead of the correct 3 192 . The most plausible explanation for this error is that she was meant to multiply a related value by 3 4 π , as it pertains to the volume of a sphere. Therefore, the correct answer is Option B: Amie should have multiplied 54 by 3 4 π .
;
