Sure, let's solve this step-by-step:
1. Find the radius of the cylinder:
- The diameter of the cylinder is given as 42 cm.
- The radius (r) is half of the diameter.
[tex]\[
r = \frac{diameter}{2} = \frac{42 \, \text{cm}}{2} = 21 \, \text{cm}
\][/tex]
2. Calculate the surface area of the cylinder:
- The surface area (A) of a closed cylinder (including both the top and bottom) is given by the formula:
[tex]\[
A = 2\pi r (r + h)
\][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height.
3. Substitute the known values into the formula:
- The radius [tex]\( (r) \)[/tex] is 21 cm.
- The height [tex]\( (h) \)[/tex] is 45 cm.
[tex]\[
A = 2\pi \cdot 21 \, \text{cm} \cdot (21 \, \text{cm} + 45 \, \text{cm})
\][/tex]
4. Simplify the expression inside the parenthesis:
[tex]\[
21 \, \text{cm} + 45 \, \text{cm} = 66 \, \text{cm}
\][/tex]
5. Calculate the surface area:
- Substitute back into the formula:
[tex]\[
A = 2\pi \cdot 21 \, \text{cm} \cdot 66 \, \text{cm} = 2\pi \cdot 1386 \, \text{cm}^2
\][/tex]
6. Calculate the numerical value of the surface area:
- Using [tex]\(\pi \approx 3.14159\)[/tex]:
[tex]\[
A \approx 2 \cdot 3.14159 \cdot 1386 \, \text{cm}^2 \approx 8708.49 \, \text{cm}^2
\][/tex]
Therefore, the surface area of the cylinder is approximately 8708.49 cm².
