To find the surface area of the pyramid using the formula you provided, we first need to clarify the components:
- \( b \) is the base of the pyramid. Since you're saying \( b \) is 3 in., we assume it's the area of the base.
- \( h \) is the height of the pyramid, which is given as 8 in.
However, in the formula \( 4(1/2)bh + b^2 \), it seems like \( b \) might also be representing the length of one side of the base if the base is a square pyramid.
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First, we calculate \( b^2 \): \[ b^2 = (3 \text{ in})^2 = 9 \text{ in}^2 \]
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Next, we'll compute the lateral surface area of the pyramid using \( 4(1/2)bh \):
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We first calculate \( (1/2)bh \): \[ (1/2)bh = (1/2)(3 \text{ in})(8 \text{ in}) = (1/2)(24 \text{ in}^2) = 12 \text{ in}^2 \]
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Now, we multiply by 4 to find the total lateral area: \[ 4 \cdot 12 \text{ in}^2 = 48 \text{ in}^2 \]
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Finally, we add the base area to the lateral surface area: \[ \text{Total Area} = \text{Lateral Area} + \text{Base Area} = 48 \text{ in}^2 + 9 \text{ in}^2 = 57 \text{ in}^2 \]
So, Leonard will need a total of 57 square inches of cardboard for the pyramid.