Question

[TON OF POINTS] [3 QUESTIONS - Easy money :)]

Answer 1

The correct answers are:

1/6πd³;
3; and 
9 cm.

Explanation:

For the first question:

The formula for the volume of a sphere is:

$V=\frac{4}{3}\pi r^3$

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:

$V=\frac{4}{3}\pi (\frac{d}{2})^3$

When we raise a fraction to a power, we raise both the numerator and the denominator to that power.  This gives us:

$V=\frac{4}{3}\pi(\frac{d^3}{2^3}) \\ \\=\frac{4}{3}\pi(\frac{d^3}{8})$

We start multiplying this:
$V=\frac{4\pi}{3}(\frac{d^3}{8}) \\ \\=\frac{4\pi d^3}{3\times 8} \\ \\=\frac{4\pi d^3}{24} \\ \\=\frac{4}{24}\pi d^3 \\ \\=\frac{1}{6}\pi d^3$

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:

$V=\frac{1}{3}(\frac{1}{2}bh)H$,

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:

$90 = \frac{1}{3}(\frac{1}{2}\times 12\times 5)H \\ \\90 = \frac{1}{3}(\frac{1}{2}\times 60)H \\ \\90 = \frac{1}{3}(30)H \\ \\90 = 10H$

We divide both sides by 10:

90/10 = 10H/10
9 = H