Question

find the distance from the point (8, -5, -7) to the origin.write your answer as a whole number or as a decimal rounded to the nearest hundredth.

Explanation:

Step1: Recall 3D distance formula

The distance \( d \) between a point \( (x, y, z) \) and the origin \( (0, 0, 0) \) is given by \( d=\sqrt{(x - 0)^2+(y - 0)^2+(z - 0)^2}=\sqrt{x^{2}+y^{2}+z^{2}} \).
For the point \( (8,-5,-7) \), we substitute \( x = 8 \), \( y=-5 \), and \( z = -7 \) into the formula. So we get \( d=\sqrt{8^{2}+(-5)^{2}+(-7)^{2}} \).

Step2: Calculate the squares

First, calculate each square: \( 8^{2}=64 \), \( (-5)^{2}=25 \), \( (-7)^{2}=49 \). Then substitute these values back into the formula: \( d=\sqrt{64 + 25+49} \).

Step3: Sum the terms inside the square root

Sum the numbers inside the square root: \( 64+25 + 49=138 \). So now we have \( d=\sqrt{138} \).

Step4: Calculate the square root and round

Calculate the square root of 138. \( \sqrt{138}\approx11.75 \) (rounded to the nearest hundredth).

Answer:

11.75