10.04 Cones and pyramids

We can find the volume of the cone using the formula V=\dfrac{1}{3}\pi r^{2}h.

We are given the radius, r=2, and height, h=6. We can substitute these into the formula to find the volume of the cone.

Calculate the volume of the original cone, then take half the volume. We need the height of the cone, so use the Pythagorean theorem with the height of the slanted lateral face as the hypotenuse and the radius of the base as the length of the triangle's short leg.

First, we'll find the height of the cone:

The height of the cone is \sqrt{96 } \text{ cm}.

Now, we need the area of the base of the original cone. We know that the radius of the base is 5 \text{ cm}, so the area of the base is \pi \left(5 ^{2} \right) = 25 \pi \text{ cm}^{2}.

We can calculate the volume of the original cone as follows:

Finally, since the cone was sawed in half, we can take half of the total cone of the original cone to be approximately 128.25 \text{ cm} ^{3}.

The volume of a cone is found by V=\dfrac{1}{3}\pi r^{2}h.

Since the perpendicular height and radius are the same, the volume is equal to:

We are given the volume is 160 \text{ mm}^{3}, so we can substitute this into our equation and solve for r.

Solving for r, we have

Let's check our answer by finding the volume of the cone using 5.35 \text{ mm} for the radius and height:

Notice that this volume is slightly differently from the given volume of 160 \text{ mm}^{3}. This is because we rounded r to two decimal places. Rounding can introduce slight differences in values, especially those involving \pi and cube roots. It's important to consider the effects of rounding when stating or checking answers.

We can set up an equation given the following:

• The volume of a cone is given by V=\dfrac{1}{3} \pi r^{2}h.

• The height of both cones remains the same.

• The new cone has 2 times the volume of the original cone.

Let's denote the radius of the original cone as r and the new cone as r_{\text{new}}.

Let the volume of the original cone be V=\dfrac{1}{3}\pi r^{2}h.

Let the volume of the new cone be V_{\text{new}}=\dfrac{1}{3}\pi r_{\text{new}}^{2}h.

Since the new cone has twice the volume of the original cone, we can say V_{\text{new}}=2\cdot\left(\dfrac{1}{3}\pi r^{2}h\right). We can use the expressions for V_{\text{new}} to write and solve an equation.

So, the length of the radius of the new cone is \sqrt{2} times the length of the radius of the original cone.

To verify that r_{\text{new}}=\sqrt{2} \cdot r, we can calculate the volumes of both cones and see if the new cone's volume is twice of the original cone's volume.

As an example, let's set r=1 \text{ in}, h=2 \text{ in}, and \pi=3.14.

Let's first calculate the volume of the original cone:

Let's calculate the volume of the new cone. Note that the radius of the new cone is \sqrt{2} times the length of the radius of the original cone:

\dfrac{\text{Volume of the new cone}}{\text{Volume of the original cone}}=\dfrac{4.19 \text{ in}^{3}} { 2.09\text{ in}^{3} } \approx 2

This verifies that the new cone's radius is indeed 2 times the original. The slight discrepancy is due to rounding.

First we find the area of the base, then we can use that to find the volume. Since this solid is a square pyramid, the base is a square.

Finding the area of the base, we have:

We can now use this to calculate the volume:

So the volume of the Pyramid of Giza is 18 \, 069 \, 333 \dfrac{1}{3} cubic Egyptian royal cubits.

Subtract the volume of the smaller pyramid from the volume of the larger pyramid.

Start by calculating the volume of the larger pyramid. We have

The volume of the larger pyramid is exactly \dfrac{512}{3} \text{ cm}^3.

Now we can calculate the volume of the smaller pyramid. We have

The volume of the smaller pyramid is exactly \dfrac{64}{3} \text{ cm}^3.

The volume of the solid after removing the smaller pyramid is \dfrac{512}{3} \text{ cm}^3 - \dfrac{64}{3} \text{ cm}^{3} = \dfrac{448}{3} \text{ cm}^{3}.

We could choose to leave the solution in exact form, since the directions do not specify rounding requirements, or we could simplify the solution and round to a measure we find appropriate.

To determine the effect on the volume, recall that the volume of a triangular pyramid is given by V = \dfrac{1}{3}BH, where B is the area of the triangular base and H is the perpendicular height.

We can calculate B using the triangle area formula A=\dfrac{1}{2}bh, where b is the base and h is the height of the triangle.

We can determine the effect on the volume by tripling the perpendicular height and the base side, calculating the new volume, then dividing the new volume by the original volume.

We are given the following:

• Perpendicular height of the pyramid \left(H\right)=24 \text{ in}

• Base side \left(b\right) = 20\text{ in}

• Height of the triangular base \left(h\right) = 17.3\text{ in}

Let's caclulate the original volume:

Let's caclulate the new volume using the tripled dimensions:

• Perpendicular height of the pyramid \left(H\right)=72 \text{ in}

• Base side \left(b\right) = 60\text{ in}

• Height of the triangular base \left(h\right) = 51.9\text{ in}

The effect of on the volume is:

\frac{V_{\text{new}}}{V_{\text{original}}}=\frac{37\,368\text{ in}^{3}}{1384\text{ in}^{3}}=27

Scaling the pyramid's height and base side length by a factor of 3 causes the volume to increase by a factor of 27.

Tripling the height and base of the original pyramid, a, results in a new pyramid, b, that maintains proportional dimensions with the original. These two pyramids are similar figures, and the ratio between their volumes is a^{3}:b^{3}.

So, increasing the dimensions of the orignal triangle by a factor of 3 results in the volume being multiplied by 3^{3} or 27.

Calculate the slant height of the cone using the Pythagorean theorem, then use the formula for the surface area of a cone to calculate the surface area.

The slant height of the cone can be found using the Pythagorean theorem as follows:

The slant height of the cone is \sqrt{90} \text{ cm}.

Now, we can calculate the surface area as follows:

The surface area of the cone is 37.46 \pi \text{ cm}^2.

We will need the height of a lateral face of the pyramid to calculate the lateral surface area. First, find the height of one lateral face using the Pythagorean theorem, then calculate the surface area of the pyramid. The diagram that follows shows the dimensions of the Pyramid of Giza:

First, we'll use the height and half the length of the base as the legs in the Pythagorean theorem:

The height of one lateral face of the pyramid is exactly \sqrt{126\,800} Egyptian royal cubits.

We can use this to calculate the area of one of the faces of the Pyramid of Giza as follows:

The area of one of the faces of the Pyramid of Giza is \left(220 \right) \left(\sqrt{126\, 800} \right)\text{ Egyptian royal cubits}^{2}, so the area of all four lateral faces is \left(880 \right) \left(\sqrt{126\,800} \right) \text{ Egyptian royal cubits}^{2} .

The surface area of the exposed walls of Pyramid of Giza to the nearest square royal cubit is 313\,359 \text{ Egyptian royal cubits}^2.

The actual length of an Egyptian royal cubit is about 20 \text{ in}, consider how the surface area compares to area of a football field. Note that we did not simplify the square root until the final calculation in order to preserve the precision of the length.

To find the new total surface area of the cone, we first calculate the slant height using the Pythagorean theorem, given the new height and the radius.

Remember that the decorative lampshade shaped as a right circular cone has no base. We only need to calculate the lateral surface area A_{\text{lateral}}=\pi r l, where r is the radius and l is the slant height.

The new height of the cone is: 8\text{ in} + 6\text{ in} = 14\text{ in}

The slant height l of the cone is

l = \sqrt{\left(14\right)^{2} + \left(4\right)^{2}} = \sqrt{196 + 16} = \sqrt{212}

Using \pi \approx 3.14, calculate the lateral surface area: