MATH PROBLEM #1

ROOM 1

1. Explain how to prove that square root of two is irrational number!

We will prove that square root of two is irrational number. square root of two is irrational number because there are no fraction numbers which can describe square root of two. We can prove that square root of two is irrational number by using Babylonian method.

two is bigger than one square and less than two square. 1²…….2…….2²

If 1² is “a”, 2 is x, and 2² is “b”, we can calculate square root of two with this equation.

square root of two Is Equal to 1,412. There is no fraction numbers which can describe square root of two.

2. Explain how to show or to indicate that the sum angle of triangle is equal to 180!

There is a right triangle which has 4cm high and 3cm in width of the base.

We can find CB,
CB equals square root of four square plus three square in bracket. So, CB equals 5 cm.

Then we can find β and γ by using trigonometry.

• Sin β is equal to four fifth. So, β is equal to arc sin four fifth. β is equal to 53.
• Sin γ is equal to three fifth. Then γ equals arc sin three fifth. So, γ is equal to 37 .

α plus β plus γ equal to 90+ 53+ 37. It is equal to 180.
So the sum angle of triangle is equal to 180 is right.

3. Explain how you are able to get phi!

We can find phi by using the area of a circle. There is a circle which has a radius x cm and it will be cut into many partitions. Then it is shaped like this:

We can find its width by using a ruler. Example the width is y cm. We can find the area by using the area of a rectangle. The area is x times y equal to phi times x square.

So, y equal to phi times x and then phi is y divided by x.

4. Find the area bounded by y = x square and y = x plus 2!

The curves are shown in this picture.

Firstly, find the intersection points of the curves by using substitution.

y equals y

x square equal to x plus two, then x square mines x mines two equal to zero.

x mines two in bracket times x plus one in bracket.

x equal to two or x equal to negative one.

We can find the area by using integral. The area is

And the area is a half.

5. Explain how you are able to determine the intersection point between the circle x square plus y square equal to twenty and y = x plus one.

Firstly we substitute y = x plus one to the circle.

x square plus open bracket x plus one close bracket square equal to twenty.

x square plus x square plus two times x plus one equal to twenty.

Two times x square plus two times x mines nineteen equal to zero.

The solution can be calculated by using abc formula.

Then substitute x₁ and x₂ into y = x + 1

y (2,622) = 2,622 + 1 = 3,622

y (-3,622) = -3,622 + 1 = -2,622

So, there are two intersection points between x square plus y square equal to twenty and y equal to x plus one. That points are ( 2,622 , 3,622 ) and ( -3,622 , -2,622 ).

Room II

1. Right Prism

Definition of right prism

A right prism is a prism in which the joining edges and faces are perpendicular to the base faces. The joining faces are rectangular. A parallelepiped is a prism of which the base is a parallelogram. A right rectangular prism is called a cuboid.

Volume

The volume of a prism is the product of the area of the base and the distance between the two base faces, or the height. The volume of a parallelepiped is the product of the area of the base and the height where height is the perpendicular distance between the base and the opposite face.

Surface Area

The surface area of a right prism is the product of two times the area of the base and the perimeter of the base times the height.

Symmetry

A right n-sided prism has two kinds of symmetry, symmetry group of a right n-sided prism and rotation symmetry.

Example Question

1. There is a right rectangular prism, which has four centimeters in height, six centimeters in width, and eight centimeters in length. Find its volume...!

Solution:

The volume of a right rectangular prism is the product of the area of the base and the height of the prism. Therefore, the volume equals four times six times eight. The volume of this prism is one hundred ninety two centimeters cube.

2. There is a right rectangular prism where the height equals the width plus two or the length plus four. This prism has eight hundred fifty six meters cube in the surface area. Find its height, its length, and its width…!

Solution:

Firstly, let the height is x meters.

The surface area of this prism is the product of two times the area of the base and the perimeter of the base times the height. Eight hundred fifty six equals two times its length times its width plus two times open bracket the length plus the width close bracket times the height. It is equals two times x plus four in bracket times x plus two in bracket plus two times open bracket two times x plus six close bracket times x. Then, eight hundred fifty six equals two times x square plus twelve times x plus sixteen plus four times x square plus twelve times x. six times x square plus twenty. Four times x minus eight hundred forty equals zero. X square plus four times x minus one hundred forty equals zero. X equals negative fourteen or x equals ten. Therefore, the height equals ten meters, the length equals fourteen meters, and the width equals twelve meters.

2. Combinatorics

Combinatory consist of multiply, permutation, and combination. We will discus about permutation and combination in this section. Permutation and combination are used to find how many ways in an event. Before we discus about it, we will discus about factorial.

Factorial

Factorial is a product of all positive integers less than or equal to n. let n is a non-negative integers. For example:

One factorial is one.

Two factorials equal to one times two.

Three factorials equal to one times two times three.

Four factorials equal to one times two times three times four.

Five factorials is one times two times three times four times five.

Permutation

A permutation is an ordered sequence of elements selected from a given finite set, without repetitions. For example, there are three elements, A, B, C; there are six sequences of elements. It is ABC, ACB, CAB, BAC, BCA, and CBA. It equals three factorials. We call permutation of three from three. So, permutation of k element from n element is n factorial over n minus k in bracket factorial

Example question:

There are three boys and four girls. They seat in a bus station.

a. How many sequences of three boys and four girls seating in the bus station

b. There are …. Sequences of three boys and four girls if four girls seat

Combination

Combination is an ordered sequence of elements selected from a given finite set, without repetitions and. ABC equal to ACB, CAB, BAC, BCA, and CBA in combination. Each k element has the same sequences k factorial. So, combination of k element from n element is n factorial divided by n minus k in bracket factorial times k factorial.

Example question: