Question

two identical rectangular prisms and a cube are joined. answer the questions to find the new solids surface area. there are square faces on the surface of the figure. each square face has an area of ft². so, the area of all the visible square faces is ft². there are non - square rectangles on the surface of the figure. each rectangle has an area of ft². so, the surface area of all the visible non - square rectangles is ft². the surface area of the figure is ft².

Explanation:

Step1: Count square faces

There are \( 4 \) square faces on the surface of the figure.

Step2: Calculate area of one square face

Since the side - length of the square (from the cube part, where the side - length of the cube is \( 2\) ft), using the formula \(A = s^2\) (where \(s = 2\) ft), \(A=2\times2 = 4\) \(ft^{2}\).

Step3: Calculate total area of square faces

Using the formula \(A_{total - square}=n\times A\) (where \(n = 4\) and \(A = 4\) \(ft^{2}\)), \(A_{total - square}=4\times4=16\) \(ft^{2}\).

Step4: Count non - square rectangles

There are \( 8 \) non - square rectangles on the surface of the figure.

Step5: Calculate area of one non - square rectangle

There are two types of non - square rectangles.

• For rectangles with dimensions \(5\times2\): \(A_1 = 5\times2=10\) \(ft^{2}\). There are \(4\) such rectangles.
• For rectangles with dimensions \(5\times2\) (another orientation) and \(2\times2\) (but we already accounted for square faces). Wait, re - checking: The non - square rectangles:
• The long rectangles: There are \(4\) rectangles with dimensions \(5\times2\) and \(4\) rectangles with dimensions \(2\times5\) (total \(8\) non - square rectangles). Using \(A = l\times w\), for \(l = 5\) ft and \(w = 2\) ft, \(A=5\times2 = 10\) \(ft^{2}\) per non - square rectangle.

Step6: Calculate total area of non - square rectangles

Using \(A_{total - non - square}=n\times A\) (where \(n = 8\) and \(A = 10\) \(ft^{2}\)), \(A_{total - non - square}=8\times10 = 80\) \(ft^{2}\).

Step7: Calculate total surface area

Using \(A_{total}=A_{total - square}+A_{total - non - square}\), \(A_{total}=16 + 80=96\) \(ft^{2}\).

Answer:

• There are \(4\) square faces on the surface of the figure.
• Each square face has an area of \(4\) \(ft^{2}\).
• So, the area of all the visible square faces is \(16\) \(ft^{2}\).
• There are \(8\) non - square rectangles on the surface of the figure.
• Each rectangle has an area of \(10\) \(ft^{2}\).
• So, the surface area of all the visible non - square rectangles is \(80\) \(ft^{2}\).
• The surface area of the figure is \(96\) \(ft^{2}\).