Question

1. rectangular prism a measures 3 inches by 4 inches by 8 inches. rectangular prism b measures 5 inches by 5 inches by 6 inches.

a. before doing any calculations, predict which prism has greater surface area to volume ratio.

b. calculate the surface area, volume, and surface area to volume ratio for each prism.

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Explanation:

Part a

• Shape - Volume Relationship:
• Prism \(A\) has more "extreme" side - length differences (3, 4, 8) compared to prism \(B\) (5, 5, 6). Generally, more "elongated" or irregular - shaped prisms (in terms of side - length ratios) tend to have a larger surface - area - to - volume ratio. So, predict that prism \(A\) has a greater surface - area - to - volume ratio.

Part b

For Prism \(A\) with \(l = 3\) in, \(w = 4\) in, \(h = 8\) in

• Surface Area (\(SA\)):
• The formula for the surface area of a rectangular prism is \(SA=2(lw + lh+wh)\).
• Substitute \(l = 3\), \(w = 4\), \(h = 8\) into the formula:
• \(SA=2(3\times4 + 3\times8+4\times8)\)
• First, calculate the products inside the parentheses: \(3\times4 = 12\), \(3\times8 = 24\), \(4\times8 = 32\).
• Then, sum them up: \(12 + 24+32=68\).
• Multiply by 2: \(SA = 2\times68=136\) square inches.
• Volume (\(V\)):
• The formula for the volume of a rectangular prism is \(V=lwh\).
• Substitute \(l = 3\), \(w = 4\), \(h = 8\): \(V=3\times4\times8=96\) cubic inches.
• Surface - Area - to - Volume Ratio (\(r_A\)):
• The ratio \(r_A=\frac{SA}{V}\).
• Substitute \(SA = 136\) and \(V = 96\): \(r_A=\frac{136}{96}=\frac{17}{12}\approx1.42\).

For Prism \(B\) with \(l = 5\) in, \(w = 5\) in, \(h = 6\) in

• Surface Area (\(SA\)):
• Use the formula \(SA = 2(lw+lh + wh)\).
• Substitute \(l = 5\), \(w = 5\), \(h = 6\): \(SA=2(5\times5 + 5\times6+5\times6)\).
• Calculate the products: \(5\times5 = 25\), \(5\times6 = 30\), \(5\times6 = 30\).
• Sum them up: \(25 + 30+30 = 85\).
• Multiply by 2: \(SA=2\times85 = 170\) square inches.
• Volume (\(V\)):
• Use the formula \(V=lwh\).
• Substitute \(l = 5\), \(w = 5\), \(h = 6\): \(V=5\times5\times6=150\) cubic inches.
• Surface - Area - to - Volume Ratio (\(r_B\)):
• The ratio \(r_B=\frac{SA}{V}\).
• Substitute \(SA = 170\) and \(V = 150\): \(r_B=\frac{170}{150}=\frac{17}{15}\approx1.13\).

Answer:

Part a

Predict that prism \(A\) has a greater surface - area - to - volume ratio.

Part b

• Prism \(A\):
• Surface area: \(136\) square inches.
• Volume: \(96\) cubic inches.
• Surface - area - to - volume ratio: \(\frac{17}{12}\approx1.42\).
• Prism \(B\):
• Surface area: \(170\) square inches.
• Volume: \(150\) cubic inches.
• Surface - area - to - volume ratio: \(\frac{17}{15}\approx1.13\).