Question

a box has dimensions 3 ft by 4 ft by 12 ft. what is the length of its space diagonal?
a. 12 ft
b. 13.41 ft
c. 14.24 ft
d. 13 ft

a point a is located at (1, 1, 1) and point b is at (4, 5, 9). what is the distance between these points?
a. 10.00
b. 9.43
c. 10.33
d. 9.11

Explanation:

Step1: Recall space - diagonal formula

For a rectangular box with length $l$, width $w$, and height $h$, the space - diagonal $d$ is given by $d=\sqrt{l^{2}+w^{2}+h^{2}}$. Here, $l = 3$ ft, $w = 4$ ft, and $h=12$ ft. So $d=\sqrt{3^{2}+4^{2}+12^{2}}$.

Step2: Calculate the squares

$3^{2}=9$, $4^{2}=16$, and $12^{2}=144$. Then $3^{2}+4^{2}+12^{2}=9 + 16+144=169$.

Step3: Find the square - root

$d=\sqrt{169}=13$ ft.

Step4: Recall 3 - D distance formula

The distance $D$ between two points $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ in 3 - D space is $D=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}+(z_2 - z_1)^{2}}$. Here, $x_1 = 1,y_1 = 1,z_1 = 1,x_2 = 4,y_2 = 5,z_2 = 9$. So $D=\sqrt{(4 - 1)^{2}+(5 - 1)^{2}+(9 - 1)^{2}}$.

Step5: Calculate the differences squared

$(4 - 1)^{2}=9$, $(5 - 1)^{2}=16$, $(9 - 1)^{2}=64$. Then $(4 - 1)^{2}+(5 - 1)^{2}+(9 - 1)^{2}=9+16 + 64=89$.

Step6: Find the square - root

$D=\sqrt{89}\approx9.43$.

Answer:

1. d. 13 ft
2. b. 9.43