When the length of one of the sides is multiplied by a constant the effect is to multiply the original volume by that constant, as for the example in
[link] .
Right pyramids, right cones and spheres
A pyramid is a geometric solid that has a polygon base and the base is joined to a point, called the apex. Two examples of pyramids are shown in the left-most and centre figures in
[link] . The right-most figure has an apex which is joined to a circular base and this type of geometric solid is called a cone. Cones are similar to pyramids except that their bases are circles instead of polygons.
Surface Area of a Pyramid
The surface area of a pyramid is calculated by adding the area of each face together.
If a cone has a height of
$h$ and a base of radius
$r$ , show that the surface area is
$\pi {r}^{2}+\pi r\sqrt{{r}^{2}+{h}^{2}}$ .
The cone has two faces: the base and the walls. The base is a circle of radius
$r$ and the walls can be opened out to a sector of a circle.
This curved surface can be cut into many thin triangles with height close to
$a$ (
$a$ is called the
slant height ). The area of these triangles will add up to
$\frac{1}{2}\times $ base
$\times $ height(of a small triangle) which is
$\frac{1}{2}\times 2\pi r\times a=\pi ra$
$a$ can be calculated by using the Theorem of Pythagoras. Therefore:
A triangular pyramid is placed on top of a triangular prism. The prism has an equilateral triangle of side length 20 cm as a base, and has a height of 42 cm. The pyramid has a height of 12 cm.
Find the total volume of the object.
Find the area of each face of the pyramid.
Find the total surface area of the object.
We use the formula for the volume of a triangular prism:
To find the total surface area, we must subtract the area of one face of the pyramid from the area of the prism. We must also subtract the area of one of the triangular faces of the prism. Doing this gives the total surface area as:
$1120-420+1680-420=1960$ This is the answer to part c.
Calculate the volumes and surface areas of the following solids: *Hint for (e): find the perpendicular height using Pythagoras.
Water covers approximately 71% of the Earth's surface. Taking the radius of the Earth to be 6378 km, what is the total area of land (area not covered by water)?
Questions & Answers
can someone help me with some logarithmic and exponential equations.
In this morden time nanotechnology used in many field .
1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc
2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc
3- Atomobile -MEMS, Coating on car etc.
and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.