As we were approaching the problem of how to design a gutter which would carry the most water from the roof with a 12 inch wide piece of material, we came up with a thesis; the length of the roof can be any number according to the roof sections, the maximum volume of the gutter will then depend on the maximum area out of the shapes' side (flank) with respect to the 12in-wide-material piece. Thus the areas of four different shapes were found. By process of elimination, the optimal area for each shape was found by deriving the equations of area as well as basic geometry. The shapes used for this project were a semi-circle, triangle, square, and rectangle. For all four shapes we maximized the area rather than the volume because the length of the gutter system does not play an important factor.

For the semi-circle, it was easy to assume that the circumference was 24 inches, considering that half of the circumference needed to be equal to 12 inches, the width of the manufactured piece of material. The equation for circumference is C=2πr. By plugging in 24 for C, we were able to find the radius, r, to be 12/π. Since finding the optimum area is needed, we could plug in the radius into the equation A=πr², the area of a circle. By substitution, we found the area to be 144/π or 45.8 in². However, this is the area of the full circle, so half of this number is equal to the semi-circle's area. Thus, 22.9 in² is the maximum amount of water a semi-circle can hold.

If we assume the shape of the gutter to be an isosceles triangle, then the angle between the two isosceles sides results in the maximum value of the area of the triangle. It makes sense for both sides, AB and AC, to be of equal length, so each side is 6 in. The diagram of the triangle is drawn below. AB + AC = 12 (12in-wide-piece)

AB = AC (isosceles triangle)
AB = 12/2 = 6in, AC = 6in also.
The...

...Name: Ernest Ng
Class: 4G (23)
Date: 2-7-06
Mathematics ACE: Ellipse Areas
Before we embark on solving the problem, let us first explore the definition of ellipse.
[pic]
An ellipse is a curve that is the locus of all points in the plane the sum of whose distances [pic] and [pic] from two fixed points [pic] and [pic] (the foci) separated by a distance of [pic] is a given positive constant [pic]
[pic]
While [pic] is called the major axis, [pic] is the semi major...

...Surface Area Formulas
In general, the surface area is the sum of all the areas of all the shapes that cover the surface of the object.
Cube | Rectangular Prism | Prism | Sphere | Cylinder | Units
Note: "ab" means "a" multiplied by "b". "a2" means "a squared", which is the same as "a" times "a".
Be careful!! Units count. Use the same units for all measurements. Examples
|Surface Area of a Cube = 6 a 2...

...Section
Question 1.
a) What number must be subtracted from 2x3 – 5x2 + 5x so that the resulting polynomial has a factor 2x – 3 ? [3]
b) D, E, F are mid points of the sides BC, CA and AB respectively of a Δ ABC. Find the ratio of the areas of Δ DEF and Δ ABC. [3]
c) A man borrowed a sum of money and agrees to pay off by paying Rs 3150 at the end of the first year and Rs 4410 at the end of the second year. If the rate of compound interest is 5%...

... = 3(5) – 3
= 12 m
(6.03)
61) What is the area of a square that has a side length of:
a) 9 units
b) 11 units
Area of a square: = (Side length)2
a) A = (9)2 b) A = (11)2
= 81 u2 = 121 u2
62) A rectangle has a length of 8 in and a width of 5 in. Find the area.
Area of a rectangle: = (L)(W)
A...

...Area of some rectilinear figures
Area of a Triangle : 1/2 x Base x Heights
Area of Equilateral Triangle : Sqrt(3)/4 x (Side)2
Area of a Triangle - Hero's formula :
Sqrt{s (s - a) (s - b) (s - c)} where 2s = a + b + c.
Area of Rectangle = Length x Breadth
Area of a Square = (Side)2
Area of Four Walls = 2 x height (Length + breadth)
Area of Rhombus = 1/2 x (Products of...

...Area 51
Area 51 is a nickname for a military base that is located in the southern portion of Nevada in the western United States (83 miles north-northwest of downtown Las Vegas). Situated at its center, on the southern shore of Groom Lake, is a large secretive military airfield. The base's primary purpose is to support development and testing of experimental aircraft and weapons systems.
Its secretive nature and undoubted connection to classified aircraft...

...Force vs. Area
Connor Blackmon
Chemistry I H, 1st Period
Mrs. Kris Clements
October 18, 2012
Problem
Will a balloon pop if it is places on a bed of nails and pressure is applied?
Hypothesis
If a balloon is placed on a bed of nails and a force is applied, then the balloon will burst.
Variables
Independent variable- Force applied to the balloon and number of nails
Dependent variable- Does the balloon burst?
Materials
14 inch by 14 in by .75 in...

...Scheduled Areas and Tribal Areas : Scheduled Tribes live in contiguous areas unlike other communities. It is, therefore, much simpler to have an area-approach for development activities and also regulatory provisions to protect their interests. In order to protect the interests of Scheduled Tribes with regard to land alienation and other social factors, provisions of ‘‘Fifth Schedule" and ‘‘Sixth Schedule"have been enshrined in the...