Space Diagonal and the Super Pythagorean Theorem
|Quantity A|Quantity B|
|---|---|
|Length of the space diagonal of a cube with sides of length 5| Length of a space diagonal of a rectangular prism with dimensions 3 by 5 by 6|
There is no need to simplify this radical, or even try to evaluate it. When you do the same process to find the value for Quantity B, it might also have a radical, so it shall be easy to compare.
Find the value for Quantity B by using the __Super Pythagorean Theorem__ with _a_ = 3, _b_ = 5, and _c_ = 6.
$$\displaystyle 3^2+5^2+6^2 = d^2$$
$$\displaystyle 9+25+36 = d^2$$
$$\displaystyle 70 = d^2$$
$$\displaystyle \sqrt{70} = d^2$$
Therefore, Quantity A is greater.
Incorrect.
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It may be tempting to think that the value of Quantity B will be greater since 6 is the length of one of the sides. However, using the __Super Pythagorean Theorem__ correctly will show a different result.
Incorrect.
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Since the dimensions of the cube and the dimensions of the rectangular prism use numbers centered around the value of 5, it may be tempting to believe that the two space diagonals would have the same length. However, this is not the case.
Incorrect.
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There is enough information to determine a relationship between the two quantities. Use the formula with the dimensions provided to get values for each of the quantities.
Well done!
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To find a value for Quantity A, use the __Super Pythagorean Theorem__ where _a_, _b_, and _c_ all have the length of 5.
$$\displaystyle 5^2+5^2+5^2 = d^2$$
$$\displaystyle 25+25+25=d^2$$
$$\displaystyle 75=d^2$$
$$\displaystyle \sqrt{75} = d$$
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.
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