Question
Ahmed found the volume of a cone having both a height and a diameter of 6 centimeters. His work is shown below.
Step 1 Find the radius.
radius = StartFraction 6 Over 2 EndFraction = 3 centimeters
Step 2 Find the area of the base.
Area of base = Pi (3) squared = 9 pi centimeters squared
Step 3 Multiply the area of the base by the height.
9 pi times 6 = 54 centimeters cubed
Step 4 Divide 54 by 3 to find the volume of the cone.
Volume of the cone = StartFraction 54 Over 3 EndFraction = 18 centimeters cubed
What was Ahmed’s first error?
In Step 1, the radius should be found by multiplying the diameter by 2.
In Step 2, the formula 2 pi r should be used to find the area of the circle.
In Step 3, the resulting area should be in terms of pi.
In Step 4, the units of the resulting volume should be Centimeters squared.
Step 1 Find the radius.
radius = StartFraction 6 Over 2 EndFraction = 3 centimeters
Step 2 Find the area of the base.
Area of base = Pi (3) squared = 9 pi centimeters squared
Step 3 Multiply the area of the base by the height.
9 pi times 6 = 54 centimeters cubed
Step 4 Divide 54 by 3 to find the volume of the cone.
Volume of the cone = StartFraction 54 Over 3 EndFraction = 18 centimeters cubed
What was Ahmed’s first error?
In Step 1, the radius should be found by multiplying the diameter by 2.
In Step 2, the formula 2 pi r should be used to find the area of the circle.
In Step 3, the resulting area should be in terms of pi.
In Step 4, the units of the resulting volume should be Centimeters squared.
Answers
Answered by
GPT-4o mini
Ahmed's first error is found in **Step 1**: "In Step 1, the radius should be found by multiplying the diameter by 2."
The correct approach is to **divide the diameter by 2** to find the radius. Since the diameter of the cone is 6 centimeters, the radius is indeed \(6 \div 2 = 3\) centimeters, which Ahmed correctly calculated. Therefore, his first error is not actually here; he calculated the radius correctly.
The other options can be analyzed as follows:
- In **Step 2**, Ahmed correctly used the formula for the area of the base of the cone, \( \pi r^2 \), where \( r = 3 \) centimeters.
- In **Step 3**, he should note that the area calculated (9π cm²) is correctly multiplied by the height (6 cm) to get \( 9π \cdot 6 = 54π \) cm³. This step should keep the volume in terms of π, rather than simplifying to 54 cm³.
- In **Step 4**, he incorrectly divided, resulting in volume in cm³ instead of expressing it as \( 18π \) cm³, which is the correct volume for the cone.
So the focus of the original query is on the error related to the calculation of volume specifically, and the important aspect to highlight is which numerical simplification was incorrect rather than the initial calculation of the radius.
Therefore, the most significant error relevant to the calculations presented, is in **Step 4**, where he incorrectly concluded that the volume was \( 18 \) cm³, rather than expressing it as \( 18\pi \) cm³.
The correct approach is to **divide the diameter by 2** to find the radius. Since the diameter of the cone is 6 centimeters, the radius is indeed \(6 \div 2 = 3\) centimeters, which Ahmed correctly calculated. Therefore, his first error is not actually here; he calculated the radius correctly.
The other options can be analyzed as follows:
- In **Step 2**, Ahmed correctly used the formula for the area of the base of the cone, \( \pi r^2 \), where \( r = 3 \) centimeters.
- In **Step 3**, he should note that the area calculated (9π cm²) is correctly multiplied by the height (6 cm) to get \( 9π \cdot 6 = 54π \) cm³. This step should keep the volume in terms of π, rather than simplifying to 54 cm³.
- In **Step 4**, he incorrectly divided, resulting in volume in cm³ instead of expressing it as \( 18π \) cm³, which is the correct volume for the cone.
So the focus of the original query is on the error related to the calculation of volume specifically, and the important aspect to highlight is which numerical simplification was incorrect rather than the initial calculation of the radius.
Therefore, the most significant error relevant to the calculations presented, is in **Step 4**, where he incorrectly concluded that the volume was \( 18 \) cm³, rather than expressing it as \( 18\pi \) cm³.
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