To find the length of the spider's thread, we will use the three-dimensional distance formula, which can be visualized as the diagonal of a rectangular prism. The formula for the diagonal \(d\) of a rectangular prism with dimensions length \(l\), width \(w\), and height \(h\) is given by:
\[ d = \sqrt{l^2 + w^2 + h^2} \]
Given the tank dimensions:
- Length \(l = 35 , \text{cm}\)
- Width \(w = 12 , \text{cm}\)
- Height \(h = 16 , \text{cm}\)
We can plug in these values into the formula.
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Calculate \(l^2, w^2,\) and \(h^2\):
- \(l^2 = 35^2 = 1225\)
- \(w^2 = 12^2 = 144\)
- \(h^2 = 16^2 = 256\)
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Sum these values: \[ l^2 + w^2 + h^2 = 1225 + 144 + 256 = 1625 \]
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Now take the square root: \[ d = \sqrt{1625} \] Calculating the square root: \[ \sqrt{1625} \approx 40.311 \]
Next, we need to convert centimeters to inches, knowing that \(1 , \text{inch} = 2.54 , \text{cm}\): \[ d , \text{in inches} = \frac{40.311 , \text{cm}}{2.54} \approx 15.87 , \text{inches} \]
Rounding to the nearest tenth, we find:
- The length of the diagonal of the bottom of the tank is approximately 15.9 inches.
- The length of the spider's thread is approximately 15.9 inches.
Final answers:
- The diagonal of the bottom of the tank is 15.9 inches.
- The length of the spider's thread is 15.9 inches.
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