To find the total surface area of the cylindrical curtain rod with the two spherical ornaments, we need to calculate the surface area of both the cylinder and the spheres, then combine these areas while accounting for any overlaps.
Step 1: Calculate the surface area of the cylinder
The formula for the surface area \( A \) of a cylinder is given by:
\[ A_{\text{cylinder}} = 2\pi r_{\text{cylinder}}h + 2\pi r_{\text{cylinder}}^2 \]
Where:
- \( r_{\text{cylinder}} \) is the radius of the cylinder,
- \( h \) is the height (or length) of the cylinder.
First, we find the radius of the cylinder:
\[ \text{Diameter of the rod} = 1.5 , \text{inches} \quad \Rightarrow \quad r_{\text{cylinder}} = \frac{1.5}{2} = 0.75 , \text{inches} \]
Next, the height of the cylinder is:
\[ h = 48 , \text{inches} \]
Now we can calculate the surface area:
\[ A_{\text{cylinder}} = 2\pi(0.75)(48) + 2\pi(0.75)^2 \]
Calculate each part:
- Side area:
\[ 2\pi(0.75)(48) = 72\pi \]
- Top and bottom area (2 bases):
\[ 2\pi(0.75)^2 = 2\pi(0.5625) = 1.125\pi \]
Adding these together:
\[ A_{\text{cylinder}} = 72\pi + 1.125\pi = 73.125\pi \]
Step 2: Calculate the surface area of the spheres
The formula for the surface area \( A \) of a sphere is given by:
\[ A_{\text{sphere}} = 4\pi r_{\text{sphere}}^2 \]
Where \( r_{\text{sphere}} \) is the radius of the sphere. The diameter of each ornament is 4 inches, so the radius is:
\[ r_{\text{sphere}} = \frac{4}{2} = 2 , \text{inches} \]
Now we calculate the surface area for one sphere and then multiply by 2 for both spheres:
\[ A_{\text{sphere}} = 4\pi(2)^2 = 4\pi(4) = 16\pi \]
For two spheres:
\[ A_{\text{spheres}} = 2 \times 16\pi = 32\pi \]
Step 3: Combine the surface areas
Now, we add the surface areas of the cylinder and the two spheres, but we need to subtract the area of the circles at the ends of the cylinder where the spheres attach. The area of each end circle is:
\[ A_{\text{end circle}} = \pi(0.75)^2 = \pi(0.5625) = 0.5625\pi \]
Since the spheres cover both ends, we subtract 2 times this area:
\[ A_{\text{ends}} = 2 \times 0.5625\pi = 1.125\pi \]
Thus the total surface area \( A_{\text{total}} \) is:
\[ A_{\text{total}} = A_{\text{cylinder}} + A_{\text{spheres}} - A_{\text{ends}} \]
Plugging in the values we found:
\[ A_{\text{total}} = 73.125\pi + 32\pi - 1.125\pi = (73.125 + 32 - 1.125)\pi = 104\pi \]
Step 4: Numerical Calculation
Now we calculate the numerical value:
\[ A_{\text{total}} = 104\pi \approx 104 \times 3.14159 \approx 326.725 \]
Rounding this to the nearest whole number, we get:
\[ \text{Total Surface Area} \approx 327 , \text{square inches} \]
Thus, the final answer is:
\[ \boxed{327} \]