Question

1. the formula for the space diagonal of a rectangular prism is: a. (l^{2}+w^{2}+h^{2}) b. (l + w+h) c. (sqrt{l^{2}-w^{2}-h^{2}}) d. (sqrt{l^{2}+w^{2}+h^{2}})

Explanation:

Step1: Recall 3 - D distance concept

In a rectangular prism with length $l$, width $w$, and height $h$, we can use the three - dimensional distance formula. If we consider the right - angled triangle formed in 3 - D space, we first find the diagonal of the base using the Pythagorean theorem: $d_{base}=\sqrt{l^{2}+w^{2}}$. Then, the space diagonal $d$ forms a right - angled triangle with the height $h$ and the base diagonal $d_{base}$.

Step2: Apply Pythagorean theorem again

Using the Pythagorean theorem on the right - angled triangle with sides $d_{base}=\sqrt{l^{2}+w^{2}}$ and $h$, we have $d^{2}=(\sqrt{l^{2}+w^{2}})^{2}+h^{2}=l^{2}+w^{2}+h^{2}$. So, $d = \sqrt{l^{2}+w^{2}+h^{2}}$.

Answer:

D. $\sqrt{l^{2}+w^{2}+h^{2}}$