The correct answers are:
1/6πd³;
3; and
9 cm.
Explanation:
For the first question:
The formula for the volume of a sphere is:
[tex]V=\frac{4}{3}\pi r^3[/tex]
We have the diameter, not the radius. The diameter is twice as much as the radius; this means to find the radius using the diameter, we divide by 2. This gives us:
[tex]V=\frac{4}{3}\pi (\frac{d}{2})^3[/tex]
When we raise a fraction to a power, we raise both the numerator and the denominator to that power. This gives us:
[tex]V=\frac{4}{3}\pi(\frac{d^3}{2^3})
\\
\\=\frac{4}{3}\pi(\frac{d^3}{8})[/tex]
We start multiplying this:
[tex]V=\frac{4\pi}{3}(\frac{d^3}{8})
\\
\\=\frac{4\pi d^3}{3\times 8}
\\
\\=\frac{4\pi d^3}{24}
\\
\\=\frac{4}{24}\pi d^3
\\
\\=\frac{1}{6}\pi d^3[/tex]
For the second question:
The planes of symmetry are the planes through which we can fold the figure in half. These are in the middle horizontally through the figure; in the middle vertically (through the width) through the figure; and in the middle vertically (through the length) through the figure. This makes 3.
For the third question:
The formula for the volume of a triangular prism is:
[tex]V=\frac{1}{3}(\frac{1}{2}bh)H[/tex],
where b is the base of the triangular base of the pyramid, h is the height of the triangular base of the pyramid, and H is the height of the pyramid.
We know the volume is 90; the base of the triangle is 12 and the height is 5:
[tex]90 = \frac{1}{3}(\frac{1}{2}\times 12\times 5)H
\\
\\90 = \frac{1}{3}(\frac{1}{2}\times 60)H
\\
\\90 = \frac{1}{3}(30)H
\\
\\90 = 10H[/tex]
We divide both sides by 10:
90/10 = 10H/10
9 = H
