The volume of a 3 dimensional solid is the amount of space it occupies.
Since the end (base) of a cylinder is a circle, the area of that circle is given by the formula:
\[area = π r^2\]
The volume of a cylinder is found my multiplying the area of one end of the cylinder by its height. So the formula of the volume of a cylinder is:
\[volume = π r^2 h\]
π is Pi, approximately 3.142
r is the radius of the circular end of the cylinder
h height of the cylinder
Our problem involves calculating the volume of Amazon Echo. The height of the Echo is 235mm while the diameter of the base is 84 mm or 8.4cm . Let's first find the area of the base.
The area of the base of Amazon Echo is:
\[area = π r^2\]
\[= 3.142 x 8.4^2\]
= 221.56 cm^2
To calculate the volume, we multiply the area with the height of the Echo, which is 235mm or 23.5cm
So, volume of Alexa Echo = 221.56 * 23.5 = 5,206.6224 cm^3
An interior designer is creating a custom coffee table for a client. The top of the table is a glass triangle that needs to balance on a single support. If the coordinates of the vertices of the triangle are at (3, 6), (5, 2), and (7, 10), at what point should the support be placed?
Are we born with the ability to tell which things are close and which are far away? Or, do we learn it? Two psychologists at Cornell University, Eleanor Gibson and Richard Walk, did an interesting experiment to answer this question.
The kitten in the photograph above is sitting on a strip in the middle of a sheet of glass.
Behind the kitten is a floor directly underneath the glass; in front of it is another floor several feet below the glass. Even when the kitten is just old enough to move about, it is afraid to move off the "cliff" to the "deep" side. Chicks and baby goats just 1 day old and human infants just old enough to crawl behave exactly the same way.
What explains this phenomenon? Surprisingly, it all about inequalities!
The mosaic above was created in the fourteenth century for a wall in Morocco. Write an expression for each of the following parts of it shown in the detail below in terms of a or b or both a and b.
1. The perimeter of the square.
2. Its area.
3. The perimeter of the other rectangle.
4. Its area.
5. The perimeter of one of the trapezoids.
6. Its area.
The perimeter is just the sum of all the sides.
The area is length times height.
3. 4a + 4b.
Add all the sides of the shaded rectangle = a + b + a + b + a + b+ a + b = 4a + 4b.
4. a^2 + 2ab.
The length of the shaded rectangle = b + a + b = a + 2b.
The height of the shaded rectangle = a.
So, the area = a x (a + 2b) = a^2 + 2ab.
5. 4a + 2b
Traversing the trapezoid we get, a + a + a + b + a + b = 4a + 2b
6. ab+ b^2
Area of a trapezoid = (width 1 + width 2)/2 x height.
width 1 = a
width 2 = b + a + b = a + 2b
height = b.
So, area = (a + a + 2b)/2 x b = (2a + 2b)/2 x b = (a + b)b = ab+ b^2
The figure above appeared in a problem on an SAT exam. All of the triangles are congruent, the area of the shaded region is 84, and the area of square ABCD is 100.
1. What is the area of one of the triangles?
2. What is the total area of the entire figure?
This tapestry was woven in Peru between 1000 and 1500 A.D. The following questions refer to the rows (numbered in red) of dark brown monkeys. Name the transformation through which the
1. third row appears to be the image of the first row.
2. second row appears to be the image of the first row.
3. fourth row appears to be the image of the third row.
4. sixth row appears to be the image of the second row.
The picture on the left is by Belgian artist Peter Raedschelders is titled Turtles Forever.
Each successive row has twice as many turtles as the row above it:
1 2 4 8 16 32 64 128 ....
Complete the following proportions:
(A) 16/32 = 32/?
(B) 2/16 = 16/?
(C) 4/64 = 64/?
1. Express the ratio 81/36 in decimal form.
2. Express the ratio 36/16 in decimal form.
3. What is the number 36 called with respect to 81 and 36?
Three vertices of a rectangle in the standard (x,y) coordinate plane have the coordinates (−2,3), (4,3) and (4,2). What are the coordinates of the fourth vertex?
Problem: Dividing a Cake
For Acute Alice's birthday, Obtuse Ollie bought a square cake, 9 inches on each side. They wanted to cut into three equal pieces.
Ollie was going to cut the cake as shown at the left below, but Alice said she wouldn't want the piece in the middle.
Alice did some figuring and cut the cake as shown at the right above. Find the area of the six triangles formed.
Chameleon Tail: The tail of the West African chameleon is coiled up in a shape similar to that of a nautilus shell.
1. What is this type of curve called?
In the figure below, AC is perpendicular to BD and angle ABC and angle EFG are right angles.
2. What is BP called with respect to triangle ABC?
3. Between which two segments is BP the geometric mean?
4. Which segment is the projection of EF on the hypotenuse of triangle EFG?
5. Between which two segments is EF the geometric mean?
2. The altitude to the hypotenuse.
3. AP and PC.
4. EP and EG.
The polygons below were created by a mathematician Waclaw Sierpinski. These are also known as fractal polygons.
A trundle wheel can be used to measure distance along the ground. The distance traveled in one revolution of the wheel is equal to the circumference of the wheel.
Given that the circumference of a trundle wheel is 3 feet, find
a) its diameter to the nearest 0.1 inch.
b) the distance that the wheel travels in making five revolutions.
A square and a rectangle have the same perimeter. The square has a length and width of 4 and the rectangle has a length of 5. What is the area of the rectangle?
The pupil of the eye controls the amount of light entering the eye. The widest that it gets, in dim light, is about 8mm in diameter. In very bright light, it gets as narrow as 2mm.
What is the approximate area of the pupil when its diameter is:
Write something about yourself. No need to be fancy, just an overview.