Question
Find the area of the circle on the top of the cylinder. Use 3.14 for โ๐.
Atop
= โ๐rยฒ
โ 3.14 ร 15.8 ร 15.8
โ 783.8696
The circle on the bottom of the cylinder is the same, so:
Abottom = Atop โ 783.8696
Find the circumference of the top circle.
C
= 2โ๐r
โ 2 ร 3.14 ร 15.8
โ 99.224
Now find the area of the curved surface. The curved surface is a rectangle. One side length is the height, and the other side length is the circumference of the circle.
Aside
= C ร h
โ 99.224h
Now add the areas to find the surface area of the cylinder.
Surface area = Atop + Abottom + Aside
3,452.9952 โ 783.8696 + 783.8696 + 99.224h
3,452.9952 โ 1,567.7392 + 99.224h
1,885.256 โ 99.224h
1,885.256 รท 99.224 โ h
19 โ h
The height of the cylinder is about 19 yards.
How do they get 1,885.256
Atop
= โ๐rยฒ
โ 3.14 ร 15.8 ร 15.8
โ 783.8696
The circle on the bottom of the cylinder is the same, so:
Abottom = Atop โ 783.8696
Find the circumference of the top circle.
C
= 2โ๐r
โ 2 ร 3.14 ร 15.8
โ 99.224
Now find the area of the curved surface. The curved surface is a rectangle. One side length is the height, and the other side length is the circumference of the circle.
Aside
= C ร h
โ 99.224h
Now add the areas to find the surface area of the cylinder.
Surface area = Atop + Abottom + Aside
3,452.9952 โ 783.8696 + 783.8696 + 99.224h
3,452.9952 โ 1,567.7392 + 99.224h
1,885.256 โ 99.224h
1,885.256 รท 99.224 โ h
19 โ h
The height of the cylinder is about 19 yards.
How do they get 1,885.256
Answers
Answered by
GPT-4o mini
To find how they arrived at the value of \( 1,885.256 \), we can break down the calculations step by step as follows:
1. **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:
\[
A_{\text{top}} + A_{\text{bottom}} \approx 783.8696 + 783.8696 = 1,567.7392
\]
2. **Circumference of the Circle**:
The circumference \( C \) of the top circle is calculated as follows:
\[
C \approx 99.224
\]
3. **Curved Surface Area**:
The area of the curved surface is given as:
\[
A_{\text{side}} = C \times h \approx 99.224h
\]
4. **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:
\[
\text{Surface Area} = A_{\text{top}} + A_{\text{bottom}} + A_{\text{side}} \approx 1,567.7392 + 99.224h
\]
5. **Given Surface Area**:
The problem states that the total surface area is approximately \( 3,452.9952 \):
\[
3,452.9952 \approx 1,567.7392 + 99.224h
\]
6. **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.
1. **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:
\[
A_{\text{top}} + A_{\text{bottom}} \approx 783.8696 + 783.8696 = 1,567.7392
\]
2. **Circumference of the Circle**:
The circumference \( C \) of the top circle is calculated as follows:
\[
C \approx 99.224
\]
3. **Curved Surface Area**:
The area of the curved surface is given as:
\[
A_{\text{side}} = C \times h \approx 99.224h
\]
4. **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:
\[
\text{Surface Area} = A_{\text{top}} + A_{\text{bottom}} + A_{\text{side}} \approx 1,567.7392 + 99.224h
\]
5. **Given Surface Area**:
The problem states that the total surface area is approximately \( 3,452.9952 \):
\[
3,452.9952 \approx 1,567.7392 + 99.224h
\]
6. **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.