Question

jen has a small wooden house that she wants to decorate. she plans to start by painting the entire surface of the house yellow. what is the area of the surface jen plans to paint? square centimeters

Explanation:

Step1: Calculate the area of the triangular parts

The formula for the area of a triangle is \(A = \frac{1}{2}bh\). Here, \(b = 4\) cm and \(h=2.24\) cm. For two triangular parts, \(A_{triangles}=2\times\frac{1}{2}\times4\times2.24 = 4\times2.24=8.96\) \(cm^{2}\)

Step2: Calculate the area of the rectangular parts on the sides of the triangular prism part

The two rectangles have dimensions \(3\times3\) and \(3\times3\). Their combined area \(A_{side - rectangles}=2\times(3\times3)=18\) \(cm^{2}\)

Step3: Calculate the area of the rectangular part on the top of the triangular prism part

The rectangle has dimensions \(4\times3\). So \(A_{top - rectangle}=4\times3 = 12\) \(cm^{2}\)

Step4: Calculate the surface area of the rectangular - prism part

The formula for the surface area of a rectangular prism is \(SA=2(lw+lh + wh)\). But we can also calculate it as follows:
The two faces with dimensions \(6\times3\): \(2\times(6\times3)=36\) \(cm^{2}\)
The two faces with dimensions \(6\times4\): \(2\times(6\times4) = 48\) \(cm^{2}\)
The bottom face (we don't paint the bottom? No, wait, the problem says "entire surface". Wait, no - actually, if we consider the combined solid. Let's re - calculate properly.

Let's consider the entire solid as a combination.
The total surface area:

• The two triangular faces: \(2\times\frac{1}{2}\times4\times2.24=8.96\)
• The three rectangular faces of the "roof" (triangular prism part):
• Two with dimensions \(3\times3\): \(2\times9 = 18\)
• One with \(4\times3\): \(12\)
• The rectangular - prism part:
• The back face (same as the front face which has the triangle + rectangle): The front face (triangle + rectangle) has area \(8.96\div2+12\div2+ (3\times4)\div2\) (no, better way:

The total surface area:

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Another way:
The figure is composed of a triangular prism on top of a rectangular prism.
For the triangular prism:

• Two triangular bases: \(A_{triangles}=2\times\frac{1}{2}\times4\times2.24 = 8.96\)
• Three rectangular lateral faces: \(3\times3 + 3\times3+4\times3=9 + 9+12 = 30\)

For the rectangular prism (excluding the area where it is attached to the triangular prism, which is \(4\times6\)):

• The total surface area of a rectangular prism \(2(lw+lh+wh)=2(6\times4 + 6\times3+4\times3)=2(24 + 18+12)=108\), but we subtract the \(4\times6\) (the attached area) and add the other parts.

The correct formula:

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Answer:

\(135\) (rounded to the nearest whole number. If we calculate \(8.96+30 + 96=134.96\approx135\))