Use the image to answer the question.

Question

An illustration shows a 3-dimensional vertical rectangular prism that appears as the letter upper C, with a rectangular portion removed from the right center. The rectangular prism is 15 feet high, 10 feet wide, and 1 foot deep. The width of the portion removed from the middle on the right side is 7 feet across. The height of the remaining two arms left at the top and bottom is 4 feet each.

In Los Cabos, Mexico, there is a big sculpture of block letters that welcomes you. The image shows the letter C
. What is its surface area?

(1 point)
__ft.2

Answer 1

To calculate the surface area of the C-shaped rectangular prism, we need to break it down into its component parts:

1. Outer Dimensions:

   • Height (h): 15 feet
   • Width (w): 10 feet
   • Depth (d): 1 foot

2. Inner Cutout:

   • Width of the cutout: 7 feet
   • Height of the remaining arms: 4 feet (with 4 feet on top and 4 feet on bottom)
   • The height of the cutout: It spans the remaining height of the rectangular prism which is \(15 - 4 - 4 = 7\) feet.

Step 1: Calculate the surface area of the outer part of the letter C

The outer surface area is made up of three rectangular components:

• Two vertical side rectangles (top and bottom arms)
• The back face

1. Front Surface Area:
   • The front consists of the main C surface which is calculated as:
     • Top rectangular portion: \( \text{Width} \times \text{Depth} = 10 \times 1 = 10 , \text{ft}^2 \)
     • Bottom rectangular portion: Same as the top, \( 10 \times 1 = 10 , \text{ft}^2 \)
     • Vertical sides: These are the left side and the right curved back:
       • Left side is: \( 15 \times 1 = 15 , \text{ft}^2 \)
       • Right side (after cutout): total height is 15, with an exclusion of the 7-foot cutout, leaving \(15 - 7 = 8\) feet; thus: \( 8 \times 1 = 8 , \text{ft}^2 \)

Total of front and sides:

• Top rectangle: \(10\)
• Bottom rectangle: \(10\)
• Left side: \(15\)
• Right side: \(8\)

Total Front area = 10 + 10 + 15 + 8 = 43 ft².

Step 2: Calculate the area of the back face

Since this is a rectangular prism with an opening:

• The back area (outer surface) remains untouched = height x depth = \(15 \times 1 = 15 \text{ ft}^2\).

Step 3: Calculate the surface area of the inner cut-out

The area of the cutout portion needs to be considered too:

• Width of the cutout: \(7\) feet
• Total height of cut (as explained above): \(7\) feet (from the cutout).

Calculate the surface of the inside of the cutout:

• Two vertical rectangles: Each side must go up to the height of the cutout (7 feet):
  • Area: Left rectangle from the left side of the C: \(4\) ft (the remaining top arm) + \(4\) ft (the bottom part which remains) = 8 feet total height to the edge of the C.
  • Each of those sides:
    • Total of two sides of cutout = \(2 \times 7\).
• Plus the inner base of the cutout:
  • The inner base of cutout: height 7 ft, width (thickness, depth) 1 ft:
  • Covered area will include the depths.

Step 4: Add all up for the total surface area

One has to add all sides, excluding overlapping areas:

• Total outer surface + the back area + internal areas of the cuts to total the overall surface area.

Calculating final result provides the total surface area of the C shape.

Final surface area turns out to be: Total Surface Area = \(43 ft^2 + 15 ft^2 + (8 + 7) ft^2 \text{(internal portions)}\).

Add these carefully to find the total result, yielding your surface area output.

Through this reasoning, conclude:

Total Surface Area = \(64 ft²\).