Answer:
[tex]38 cm^3[/tex]
Step-by-step explanation:
Given the slant height and a base side length of a right square pyramid, we can use these numbers to solve for the volume.
The formula for the volume of a square pyramid is generally known as:
[tex]V = 1/3 a^2h[/tex], where the variable 'a' represents the base side length and 'h' represents the height.
Since we are given the slant height, we can use Pythagorean's Theorem to calculate the height (non-slanted) when given a base side length by using the equation below:
[tex]a^2+b^2=c^2[/tex]
This is a special case because this is a right square pyramid, meaning we can use C squared and substitute that for the given slant height, 6.1 cm. The leg of the right triangle, which is [tex]\frac{s}{2}[/tex], takes place of 'b' which is 4.5. Now, given all of this, we can plug in our known values. Height is what we are solving.
[tex]h^2+\left(\frac{4.5}{2}\right)^2=6.1^2[/tex]
Now we will simplify and solve everything, and we get h; or the height, which equals 5.668977.
Now we can finally use the general formula to solve the volume of a square pyramid.
[tex]V = 1/3 a^2h[/tex]
Let's plug in everything now to find the volume — which we know that the height is approximately 5.66, and the base side length is 4.5.
[tex]V=\frac{1}{3}\left(4.5\right)^2\left(5.66\right)[/tex]
The volume from this formula equates to 38.205 cm³, or roughly 38 cm³, meaning the answer is B.
