To determine the length of one of the sides of the base of a right square pyramid where the height is 4.2 meters and the volume is 216 cubic meters, we can use the relationship between volume and the dimensions of the pyramid.
The formula for the volume of a right square pyramid is given by:
[tex]\[ V = \frac{1}{3} \times \text{base area} \times \text{height} \][/tex]
Let's denote the side length of the square base as [tex]\( s \)[/tex]. The area of the square base [tex]\( \text{base area} \)[/tex] is:
[tex]\[ \text{base area} = s^2 \][/tex]
So the volume formula becomes:
[tex]\[ V = \frac{1}{3} \times s^2 \times h \][/tex]
We need to solve for [tex]\( s \)[/tex].
Given:
[tex]\[ V = 216 \, \text{cubic meters} \][/tex]
[tex]\[ h = 4.2 \, \text{meters} \][/tex]
Plug these values into the volume formula:
[tex]\[ 216 = \frac{1}{3} \times s^2 \times 4.2 \][/tex]
First, simplify the equation:
[tex]\[ 216 = \frac{4.2}{3} \times s^2 \][/tex]
Calculate [tex]\( \frac{4.2}{3} \)[/tex]:
[tex]\[ \frac{4.2}{3} = 1.4 \][/tex]
Now the equation becomes:
[tex]\[ 216 = 1.4 \times s^2 \][/tex]
Solve for [tex]\( s^2 \)[/tex]:
[tex]\[ s^2 = \frac{216}{1.4} \][/tex]
Calculate [tex]\( \frac{216}{1.4} \)[/tex]:
[tex]\[ s^2 \approx 154.2857 \][/tex]
Take the square root of both sides to find [tex]\( s \)[/tex]:
[tex]\[ s \approx \sqrt{154.2857} \][/tex]
[tex]\[ s \approx 12.4 \][/tex]
Thus, the length of one of the sides of the base of the pyramid, rounded to the nearest tenth, is:
[tex]\[ \boxed{12.4 \, \text{meters}} \][/tex]
