To solve the problem, we need to determine the correct way to find the base, [tex]\( b \)[/tex], of a triangle given its area and height, and identify each student's mistake in their calculations.
### Correct Formula for the Area of a Triangle
The correct formula for the area of a triangle is given by:
[tex]\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
For a triangle with an area of 24 square centimeters and a height of 4 centimeters:
[tex]\[ 24 = \frac{1}{2} \times b \times 4 \][/tex]
To find [tex]\( b \)[/tex], we solve:
[tex]\[ 24 = 2b \][/tex]
[tex]\[ b = \frac{24}{2} \][/tex]
[tex]\[ b = 12 \][/tex]
Now, let's examine each student's equation and identify their mistakes.
### Jon's Equation: [tex]\( 24 = 4b \)[/tex]
Jon wrote:
[tex]\[ 24 = 4b \][/tex]
Jon's Mistake:
- Jon did not include the [tex]\(\frac{1}{2}\)[/tex] factor from the area formula of a triangle.
- He forgot to divide by 2 for the correct formula, where the equation should have been [tex]\( 24 = 2b \)[/tex].
- Therefore, Jon's calculation resulted in [tex]\( b = 6 \)[/tex] instead of [tex]\( b = 12 \)[/tex].
### Annie's Equation: [tex]\( b = \frac{1}{2} \cdot 24 \cdot 4 \)[/tex]
Annie wrote:
[tex]\[ b = \frac{1}{2} \times 24 \times 4 \][/tex]
Annie's Mistake:
- Annie multiplied both the area (24) and the height (4) directly by [tex]\(\frac{1}{2}\)[/tex] incorrectly.
- Instead of using the formula [tex]\( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)[/tex] to solve for [tex]\( b \)[/tex], she used an incorrect arrangement.
- Therefore, Annie's calculation resulted in [tex]\( b = 48 \)[/tex] instead of [tex]\( b = 12 \)[/tex].
### Antonio's Equation: [tex]\( 24 = 2 \cdot 4b \)[/tex]
Antonio wrote:
[tex]\[ 24 = 2 \times 4b \][/tex]
Antonio's Mistake:
- Antonio incorrectly used 2 instead of [tex]\(\frac{1}{2}\)[/tex] in the area formula.
- The correct equation should have been [tex]\( 24 = \frac{1}{2} \times 4 \times b \)[/tex].
- Therefore, Antonio's calculation resulted in [tex]\( b = 3 \)[/tex] instead of [tex]\( b = 12 \)[/tex].
### Summary of Mistakes:
- Jon's Mistake: Jon did not divide the base by 2 in the area formula.
- Annie's Mistake: Annie incorrectly multiplied all values together instead of using the correct area formula.
- Antonio's Mistake: Antonio used 2 instead of [tex]\(\frac{1}{2}\)[/tex] in his equation.
In conclusion, Jon's, Annie's, and Antonio's mistakes led them to incorrect base lengths of 6, 48, and 3 respectively, instead of the correct base length of 12.
