To find how they arrived at the value of \( 1,885.256 \), we can break down the calculations step by step as follows:
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Calculate the Area of the Top and Bottom Circles: \[ A_{\text{top}} \approx 783.8696 \] Since the area of the bottom circle is the same: \[ A_{\text{bottom}} \approx 783.8696 \]
Thus, the total area from both the top and the bottom circles is: \[ A_{\text{top}} + A_{\text{bottom}} \approx 783.8696 + 783.8696 = 1,567.7392 \]
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Circumference of the Circle: The circumference \( C \) of the top circle is calculated as follows: \[ C \approx 99.224 \]
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Curved Surface Area: The area of the curved surface is given as: \[ A_{\text{side}} = C \times h \approx 99.224h \]
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Total Surface Area: The total surface area of the cylinder is then calculated by summing the areas of the top, bottom, and the curved surface: \[ \text{Surface Area} = A_{\text{top}} + A_{\text{bottom}} + A_{\text{side}} \approx 1,567.7392 + 99.224h \]
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Given Surface Area: The problem states that the total surface area is approximately \( 3,452.9952 \): \[ 3,452.9952 \approx 1,567.7392 + 99.224h \]
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Solving for \( h \): To isolate \( h \), subtract \( 1,567.7392 \) from both sides: \[ 3,452.9952 - 1,567.7392 \approx 99.224h \] This calculation gives: \[ 1,885.256 \approx 99.224h \]
Thus, \( 1,885.256 \) is obtained from the difference between the total surface area and the area of both the top and bottom circles.
The remaining steps then solve for \( h \): \[ h \approx \frac{1,885.256}{99.224} \approx 19 \]
This confirms that the height of the cylinder is approximately 19 yards.
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