First, the measures we have are:
The diameter of the cylinder is:
we will call the diameter "d" for reference:
[tex]d=3\frac{1}{2}\text{ in}=3.5in[/tex]And we also have the height of the cylinder:
We will call this "h":
[tex]h=19in[/tex]The steps to find the surface area:
Step 1. The surface area of a cylinder is formed of two circles and one rectangle as shown in the following diagram:
Where h is the height, h=19in
r is the radius, defines as half of the diameter:
[tex]\begin{gathered} r=\frac{d}{2} \\ r=\frac{3.5in}{2} \\ r=1.75in \end{gathered}[/tex]And c is the circumference of the circle, defined as follows:
[tex]c=d\pi[/tex]Where pi is a constant: pi=3.1416
so we find the value of the circumference c:
[tex]\begin{gathered} c=(3.5\text{ in)(3.1415)} \\ c=10.995in \end{gathered}[/tex]Step 2. Calculate the area of the two circles of the area.
The formula to calculate the area of a circle is:
[tex]A=\pi\times r^2[/tex]We substitute our values:
[tex]\begin{gathered} A=3.1416\times(1.75in)^2 \\ \end{gathered}[/tex]And solving the operations:
[tex]\begin{gathered} A=3.1416\times3.0625in^2 \\ A=9.62in^2 \end{gathered}[/tex]And we need to multiply this are by 2 because there are two identical circles:
[tex]\begin{gathered} A=2\times9.62in^2 \\ A=19.24in^2 \end{gathered}[/tex]Step 3. Calculate the area of the rectangle.
To calculate that area we multiply the h by c:
[tex]A=h\times c[/tex]Substituting our h and c values:
[tex]\begin{gathered} A=19in\times10.995in \\ A=208.905in^2 \end{gathered}[/tex]Step 4. Add the area of the circles and the rectangle to find the total surface area:
[tex]\begin{gathered} A_{\text{surface}}=19.24in^2+208.905in^2 \\ A_{\text{Surface}}=228.145in^2 \end{gathered}[/tex]The closest value out of the options is: 228.16
Answer: 228.16
