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Mind Map Overal Idea Content Speed Notes Quick Coverage Polynomial A polynomial is an algebraic expression that includes constants, variables, and exponents. It is the expression in which the variables have only positive integral powers. Example 1. 4×3 + 3×2 + x +3 is a polynomial in variable x. 2. 4×2 + 3x-1 – 4… readmore
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Polynomial
A polynomial is an algebraic expression that includes constants, variables, and exponents. It is the expression in which the variables have only positive integral powers.
1. 4x3 + 3x2 + x +3 is a polynomial in variable x.
2. 4x2 + 3x-1 – 4 is not a polynomial as it has negative power.
3. 3x3/2 + 2x – 3 is not a polynomial.
2 is the constant term, which has no variable.(Scroll down till end of the page)
Polynomials are denoted by p (x), q (x), etc.
In the above polynomials, 2x2, 3y, and 2 are the terms of the polynomial.
2 and 3 are the coefficients of x2 and y, respectively.
x and y are the variables.
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A polynomial is an algebraic expression that includes constants, variables, and exponents. It is the expression in which the variables have only positive integral powers.
1. 4x3 + 3x2 + x +3 is a polynomial in variable x.
2. 4x2 + 3x-1 – 4 is not a polynomial as it has negative power.
3. 3x3/2 + 2x – 3 is not a polynomial.
If there is only one variable in the expression, then this is called the polynomial in one variable.
Types of polynomials on the basis of the number of terms
Types of polynomials on the basis of the number of degrees
The highest value of the power of the variable in the polynomial is the degree of the polynomial.
If p(x) is a polynomial, then the number ‘a’ will be the zero of the polynomial with p(a) = 0. We can find the zero of the polynomial by equating it to zero.
The given polynomial is p(x) = x – 4
To find the zero of the polynomial, we will equate it to zero.
x – 4 = 0
x = 4
p(4) = x – 4 = 4 – 4 = 0
This shows that if we place 4 in place of x, we get the value of the polynomial as zero. So 4 is the zero of this polynomial. And also, we are getting the value 4 by equating the polynomial with 0.
So 4 is the zero of the polynomial or the root of the polynomial.
The root of the polynomial is basically the x-intercept of the polynomial.
If the polynomial has one root, it will intersect the x-axis at one point only, and if it has two roots, it will intersect at two points, and so on.
Find p (1) for the polynomial p (t) = t2 – t + 1
p (1) = (1)2 – 1 + 1
= 1 – 1 + 1
= 1
We know the property of division which follows in the basic division, i.e.
Dividend = (Divisor × Quotient) + Remainder
This follows the division of polynomials.
If p(x) and g(x) are two polynomials in which the degrees of p(x) ≥ degree of g(x) and g(x) ≠ 0 are given, then we can get the q(x) and r(x) so that:
P(x) = g(x) q(x) + r(x),
where r(x) = 0 or degree of r(x) < degree of g(x).
It says that p(x) divided by g(x), gives q(x) as a quotient and r(x) as a remainder.
Let’s understand it with an example
We can see that ‘x’ is common in the above polynomial, so we can write it as
Hence, 3x2 + x + 1 and x the factors of 3x3 + x2 + x.
Divide x2 – 3x -10 by 2 + x
Step 1: Write the dividend and divisor in descending order, i.e., in the standard form. x2 – 3x -10 and x + 2
Divide the first term of the dividend with the first term of the divisor.
x2/x = x this will be the first term of the quotient.
Step 2: Now multiply the divisor by this term of the quotient and subtract it from the dividend.
Step 3: Now the remainder is our new dividend, so we will repeat the process again by dividing the dividend by the divisor.
Step 4: – (5x/x) = – 5
Step 5:
The remainder is zero.
Hence x2 – 3x – 10 = (x + 2)(x – 5) + 0
Dividend = (Divisor × Quotient) + Remainder
The Remainder Theorem says that if p(x) is any polynomial of degree greater than or equal to one and let ‘t’ be any real number and p (x) is divided by the linear polynomial x – t, then the remainder is p(t).
As we know,
P(x) = g(x) q(x) + r(x)
If p(x) is divided by (x-t) then
If x = t
P (t) = (t – t). q (t) + r = 0
To find the remainder or to check the multiple of the polynomial, we can use the remainder theorem.
What is the remainder if a4 + a3 – 2a2 + a + 1 is divided by a – 1.
P(x) = a4 + a3 – 2a2 + a + 1
To find the zero of (a – 1), we need to equate it to zero.
a -1 = 0
a = 1
p (1) = (1)4 + (1)3 – 2(1)2 + (1) + 1
= 1 + 1 – 2 + 1 + 1
= 2
So by using the remainder theorem, we can easily find the remainder after the division of the polynomial.
The factor theorem says that if p(y) is a polynomial with degree n≥1 and t is a real number, then
Check whether g(x) = x – 3 is the factor of p(x) = x3 – 4x2 + x + 6 using the factor theorem.
According to the factor theorem, if x – 3 is the factor of p(x), then p(3) = 0, as the root of x – 3 is 3.
P (3) = (3)3 – 4(3)2 + (3) + 6
= 27 – 36 + 3 + 6 = 0
Hence, g (x) is the factor of p (x).
Find the value of k, if x – 1 is a factor of p(x) = kx2 – √2x + 1
As x -1 is the factor, p(1) = 0
Factorization can be done by three methods
1. By taking out the common factor
If we have to factorise x2 –x then we can do it by taking x common.
x(x – 1) so that x and x-1 are the factors of x2 – x.
2. By grouping
ab + bc + ax + cx = (ab + bc) + (ax + cx)
= b(a + c) + x(a + c)
= (a + c)(b + x)
3. By splitting the middle term
x2 + bx + c = x2 + (p + q) + pq
= (x + p)(x + q)
This shows that we have to split the middle term in such a way that the sum of the two terms is equal to ‘b’ and the product is equal to ‘c’.
Factorize 6x2 + 17x + 5 by splitting the middle term.
If we can find two numbers p and q such that p + q = 17 and pq = 6 × 5 = 30, then we can get the factors.
Some of the factors of 30 are 1 and 30, 2 and 15, 3 and 10, 5 and 6, out of which 2 and 15 is the pair which gives p + q = 17.
6x2 + 17x + 5 =6 x2 + (2 + 15) x + 5
= 6 x2 + 2x + 15x + 5
= 2 x (3x + 1) + 5(3x + 1)
= (3x + 1) (2x + 5)
| Algebraic Identities |
| 1. (x + y)2 = x2 + 2xy + y2 |
| 2. (x – y)2 = x2 – 2xy + y2 |
| 3. (x + y) (x – y) = x2 – y2 |
| 4. (x + a) (x + b) = x2 + (a + b)x + ab |
| 5. (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx |
| 6. (x + y)3 = x3 + y3 + 3xy(x + y) = x3+ y3 + 3x2y + 3xy2 |
| 7. (x – y)3 = x3– y3 – 3xy(x – y) = x3 – y3 – 3x2y + 3xy2 |
| 8. x3 + y3 = (x + y)(x2 – xy + y2) |
| 9. x3 – y3 = (x – y)(x2 + xy + y2) |
| 10. x3 + y3 + z3 – 3xyz = (x + y + z)(x2 + y2 + z2 – xy – yz – zx) x3 + y3 + z3 = 3xyz if x + y + z = 0 |
Factorize 8x3 + 27y3 + 36x2y + 54xy2
The given expression can be written as
= (2x)3 + (3y)3 + 3(4x2) (3y) + 3(2x) (9y2)
= (2x)3 + (3y)3 + 3(2x)2(3y) + 3(2x)(3y)2
= (2x + 3y)3 (Using Identity VI)
= (2x + 3y) (2x + 3y) (2x + 3y) are the factors.
Factorize 4x2 + y2 + z2 – 4xy – 2yz + 4xz.
4x2 + y2 + z2 – 4xy – 2yz + 4xz = (2x)2 + (–y)2 + (z)2 + 2(2x) (-y)+ 2(–y)(z) + 2(2x)(z)
= [2x + (- y) + z]2 (Using Identity V)
= (2x – y + z)2 = (2x – y + z) (2x – y + z)
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Mind Map Overal Idea Content Speed Notes Quick Coverage Some objects can be charged by rubbing with otherobjects. There are two kinds of charges — positive charge and negative charge Like charges repel and unlike charges attract each other. (Scroll down till end of the page) Study Tools Audio, Visual & Digital Content Static Electric… readmore
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Some objects can be charged by rubbing with other
objects. There are two kinds of charges — positive charge and negative charge Like charges repel and unlike charges attract each other. (Scroll down till end of the page)
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Static Electric Charges: The electric charges produced due to rubbing are called static electric charges. The electric charges constitute electric current during their motion.
An electroscope may be used to detect whether a body is charged or not.
An electroscope is a deviced used to detect the presence of electric charge.
earthing: The process of transfer of charge from a charged object to the earth is called earthing.
Lightning
Lightning is a sudden, natural and violent phenomenon of creating bright flashes during a thunderstorm.
Lightning occurs due to a high-voltage electric discharge between two clouds or a cloud and the ground, or within a single cloud during a thunderstorm.
The formation of clouds involves friction between water droplets in the atmosphere.
The friction charges the particles in the atmosphere. The negative charges accumulate at the bottom of the cloud and the positive charges at the top.
As the accumulation of the charges increases, the cloud will induce positive charges on the ground nearby.
As the amount of charge increases. This causes an imbalance of charges between a region of a cloud and gorund or another cloud (or in a cloud itself). This imbalance is significant enough to break through air resistance.
Lightning releases an average of one gigajoule of energy.
Lightning strike could destroy life and property.
Protective measures are of utmost importance during lightning strikes.
Taking shelter in interiors (house or other closed places) and vehicles (closed e.g. cars) are the most preferred measures.
Lightning conductors can protect buildings from the effects of lightning.
Earth Quake
A natural phenomenon that cannot be predicted is an earthquake.
The earth consists of three major layers, called the crust, the mantle and the core.
The core is further divided into the inner core and the outer core.
The mantle consists of semi-solid material above which the crust floats.
The crust consists of oceans and continents.
The crust is divided into several parts, called tectonic plates.
The regions where one tectonic plate slides against another are referred to as fault zones, and these are the regions where an earthquake is likely to occur.
Hence, these zones are referred to as seismic zones.
The place in the interior of the earth where an earthquake occurs is the focus, and the region on the surface of the earth that is the closest to focus is likely to experience the largest damage.
This region is called the epicenter of the earthquake.
The instrument that measures the severity of an earthquake is a seismograph.
It basically consists of a drum that rolls and a pendulum with a stylus that traces the waves of an earthquake on a sheet like a graph paper.
The energy released at the focus propagates outwardly in form of waves known as seismic waves. Destructive energy of an earthquake is measured on the Richter scale.
It is a logarithmic scale, ranging from 1 to 10 for indicating the intensity of an earthquake.
The earthquake measuring 7 or more on Richter scale can cause severe damage to life and property. Protective measures for earthquake:
If you are at home:
Take shelter under a table and stay there till shaking stops.Stay away from tall and heavy objects that may fall on you. If you are in bed, do not get up. Protect your head with a pillow.
If you are outdoors: Find a clear spot, away from buildings, trees and overhead power lines. Drop to the ground. If you are in a car or a bus, do not come out. Ask the driver to drive slowly to a clear spot. Do not come out till the tremors stop. Moreover, it is advisable to make the structure of buildings simple so that they are ‘Quake Safe’.
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Mind Map Overal Idea Content Speed Notes Quick Coverage Content : (Scroll down till end of the page) Study Tools Audio, Visual & Digital Content Cuboid What is a cuboid? Parts And Their Alignment Of A Cuboid Faces The flat surfaces of a cuboid are known as its faces. A cuboid has six faces, and… readmore
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What is a cuboid?
The flat surfaces of a cuboid are known as its faces.
A cuboid has six faces, and each face is a rectangle.
These faces are arranged such that three pairs of opposite faces are parallel to each other.
The adjacent faces are perpendicular to each other (i.e., the angle between any two touching faces of a cube is right angle, 90°.
An edge is a line segment where the two surfaces of a cuboid meet.
There are 12 edges in a cuboid, where three edges meet at each vertex.
All edges form right angles with the adjacent edges and faces.
A vertex is a point where the three edges meet. Vertices is the plural of vertex.
Cuboid has eight vertices.
Diagonal of a cuboid is a line segment that joins two opposite vertices.
The cuboid has four space diagonals.
Length of the diagonal of cuboid = √(length2 + breadth2 + height2) units.
Cuboids exhibit high symmetry.
They have rotational symmetry of order 4, meaning that you can rotate them by 90 degrees about their centre and they will look the same.
It is a three-dimensional, Rectangular figure.
It has 6 faces, 12 edges, and 8 vertices.
All 6 faces are rectangles.
Each vertex meets three faces and three edges.
The edges run parallel to those parallel to it.
All angles of a cuboid are right angles.
The total surface area of a cuboid is defined as the area of its surface (Appearing face).
Imagine yourself sitting in a cuboid shaped room. You can then see the four walls around you. This denotes the lateral surface area of that room.
That is, the lateral surface area of a cuboid shaped room is the area of its four walls, excluding the ceiling and the floor.
The lateral surface area of the cuboid is the sum of areas of its square faces, excluding the area of the top and the bottom face.
So the lateral surface area of a cube = sum of areas of 4 faces = (Length ✕ Height) + (Length ✕ Height) + (Length ✕ Height) + (Breadth ✕ Height) + (Breadth✕ Height)
Derivation of Total Surface Area of a Cuboid
Since the total surface area of a cuboid (TSA) is the area of its surface.
Total surface area of a cuboid = Lateral Surface Area + Area Of Bottom Surface + Area Of Top Surface
Total surface area of a cuboid = Area Of Front Surface + Area Of Back Surface + Area Of Left Srface + Area Of Right Surface + Area Of Bottom Surface + Area Of Top Surface
Total surface area of a cuboid = Lateral Surface Area 2[Area Of Bottom Surface]
Since Area Of Top Surface = + Area Of Bottom Surface We get, Total surface area of a cuboid = Lateral Surface Area + 2[Area Of Top Surface]
TSA = (Length ✕ Height) + (Length ✕ Height) + (Length ✕ Height) + (Breadth ✕ Height) + (Breadth✕ Height) + (Length ✕ Breadth) + (Length ✕ Breadth)
TSA = 2(Length ✕ Height) + 2(Breadth ✕ Height) + 2(Length ✕ Breadth)
TSA = 2[(Length ✕ Height) + (Breadth ✕ Height) + (Length ✕ Breadth)]
The volume of a three-dimensional object can be defined as the space required for it.
Similarly, Volume of a cuboid is defined as the space required for the cuboid or the Space occupied by the cuboid.
The volume of a cuboid can be calculated using the formula, V = lbh, where,
l = length, b = breadth or width, h = height
This formula shows that the volume of a cuboid is directly proportional to its length, breadth and height.
The volume is calculated by multiplying the object’s length, breadth, and height.
Hence the volume of the cube = lbh = lenth ✕ breadth ✕ height
Similar Shapes:
Real-world Examples:
Fun Fact:
Remember that these notes provide an overview of cuboids, and there are more advanced topics and applications related to this shape in various fields of study.
A cube is a three-dimensional regular polyhedron characterised by its 6 Identical (Congruent) Squares in which 3 Pairs of them parallel.
The flat surfaces of a cube are known as its faces.
A cube has six faces, and each face is a perfect square. These faces are arranged such that three pairs of faces are parallel to each other.
The adjacent faces are perpendicular to each other (the angle between any two touching faces of a cube is right angle, 90°.
All the edges have the same length.
A cube also has 8 vertices and 12 edges.
An edge is a line segment where the two surfaces of a cube meet.
There are twelve edges in a cube, where three edges meet at each vertex.
All edges have equal length and form right angles with the adjacent edges and faces.
A vertex is a point where the three edges meet. Vertices is the plural of vertex.
Cube has eight vertices.
The cube has four space diagonals that connect opposite vertices, each of which has a length of √3 times the length of an edge.
Cubes exhibit high symmetry.
They have rotational symmetry of order 4, meaning that you can rotate them by 90 degrees about their centre and they will look the same.
It is a three-dimensional, square-shaped figure.
It has 6 faces, 12 edges, and 8 vertices.
All 6 faces are squares with equal area.
All sides have the same length.
Each vertex meets three faces and three edges.
The edges run parallel to those parallel to it.
All angles of a cube are right angles.
The total surface area of a cube is defined as the area of its outer surface.
Derivation of Total Surface Area of a Cube
Since the total surface area of a cube is the area of its outer surface.
total surface area of a cube = 6 ✕ area of one face.
We know that the cube has six square faces and each of the square faces is of the same size, the total surface area of a cube = 6 ✕ area of one face.
Let the length of each edge is “s”.
Area of one square face = length of edge ✕ length of edge
Area of one square face == s ✕ s = s²
Therefore, the total surface area of the cube = 6s²
The total surface area of the cube will be equal to the sum of all six faces of the cube.
Imagine yourself sitting in a cube shaped room. You can then see the four walls around you. This denotes the lateral surface area of that room.
That is, the lateral surface area of a cube shaped room is the area of its four walls, excluding the ceiling and the floor.
The lateral surface area of the cube is the sum of areas of its square faces, excluding the area of the top and the bottom face.
So the lateral surface area of a cube = sum of areas of 4 faces = 4a²
The Volume of a Cube
The volume of a three-dimensional object can be defined as the space required for it.
Similarly, Volume of a cube is defined as the space required for the cube or the Space occupied by the cube.
The volume of a cube can be calculated using the formula V = s3, where “s” represents the length of one side of the cube.
This formula shows that the volume of a cube is directly proportional to the cube of its side length.
The volume is calculated by multiplying the object’s length, breadth, and height. In the case of a cube shape, the length, width, and height are all of the same length. Let us refer to it as “s”.
Hence the volume of the cube is s ✕ s ✕ s = s³
We encounter many cubes in our daily life such as Ice cubes, sugar cubes, dice and the building blocks used in games.
Cubes play a fundamental role in the study of geometry and serve as a basis for understanding three-dimensional space and concepts such as volume and surface area.
Also, Cubes have many applications in mathematics, engineering, architecture and art etc.
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Cylinder is an important topic in Mathematics. It is a three-dimensional solid shape that has two parallel circular bases connected by a curved surface.
In this post, we will explore the properties of a cylinder and how to calculate its volume and surface area.
Let’s start with the basic definition of a right circular cylinder.
A cylinder is a solid shape that has two parallel circular bases of equal size and shape.
The curved surface that connects the two bases is called the lateral surface.
The axis of the cylinder is a line passing through the center of both bases.
(i). Solid Cylinder
(ii). Hollow Cylinder
Total Surface Area of Right Circular Cylinder = Curved Surface Area + Cicular Base Area + Circular Top Surface Area.
Since the Cicular Base Area And The Circular Top Surface Area of a cylinder are equal
Total Surface Area of Right Circular Cylinder = Curved Surface Area + 2(Cicular Top Surface Area)
Or
Total Surface Area of Right Circular Cylinder = Curved Surface Area + 2(Cicular Base Surface Area)
Therefore the formula to calculate the surface area of a cylinder is expressed as the following:
$$SA = 2\pi r h + 2\pi r^2$$
where SA is the surface area, r is the radius of the base, and h is the height of the cylinder.
Here, Curved Surface Area, $$CSA = 2\pi r h$$ and
Cicular Base Area = Top Surface Area = $$\pi r^2$$
The formula to calculate the volume of a cylinder is given by:
$$V = \pi r^2 h$$
where V is the volume, r is the radius of the base, and h is the height of the cylinder.
Now let’s take an example to understand how to use these formulas. Suppose we have a cylinder with a radius of 4 cm and a height of 10 cm. To calculate its volume, we can use the formula:
$$V = \pi (4)^2 (10) = 160\pi$$
Therefore, the volume of the cylinder is 160π cubic cm.
To calculate its surface area, we can use the formula:
$$SA = 2\pi (4) (10) + 2\pi (4)^2 = 120\pi$$
Therefore, the surface area of the cylinder is 120π square cm.
In conclusion, understanding the properties of a cylinder and how to calculate its volume and surface area is important in CBSE Class 10 Mathematics. By using the formulas mentioned above, you can easily solve problems related to cylinders.
A hollow cylinder is a three-dimensional object with a circular base and a cylindrical shape. It is also known as a cylindrical shell. The cylinder has two circular faces and a curved surface. The thickness of the cylinder is uniform and it is hollow from inside.
The volume of a hollow cylinder can be calculated using the formula V = πh(R2-r2), where h is the height of the cylinder, R is the radius of the outer circle, and r is the radius of the inner circle.
The surface area of a hollow cylinder can be calculated using the formula A = 2πh(R+r), where h is the height of the cylinder, R is the radius of the outer circle, and r is the radius of the inner circle.
Hollow cylinders are used in various applications such as pipes, drums, and containers. They are also used in engineering structures such as bridges and towers.
In conclusion, a hollow cylinder is a useful shape in various fields and can be easily calculated using mathematical equations.
A cone is a three-dimensional geometric shape that has a circular base and a single vertex. It can be visualized as a pyramid with a circular base. In this note, we will cover the basic concepts and formulas related to cones.
The surface area of a cone is the sum of the areas of its base and lateral surface. The formula to calculate the surface area of a cone is:
$$A = \pi r (r + l)$$
Where:
The volume of a cone is the amount of space enclosed by it. The formula to calculate the volume of a cone is:
$$V = \frac{1}{3} \pi r^2 h$$
Where:
Here are a few example equations related to cones:
Solid Sphere
Introduction – A solid sphere is a three-dimensional geometric figure in which all points inside the sphere are at the same distance from its center. – It is a type of 3D shape known as a “sphere” with a uniform density throughout.
Characteristics – The solid sphere has a well-defined volume, surface area, and mass. – It is completely filled with matter.
Volume of Solid Sphere The formula to calculate the volume (\(V_s\)) of a solid sphere is given by: \[ V_s = \frac{4}{3} \pi r^3 \] Where: \(V_s\) = Volume of the solid sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(r\) = Radius of the sphere.
Surface Area of Solid Sphere The formula to calculate the surface area (\(A_s\)) of a solid sphere is given by: \[ A_s = 4 \pi r^2 \]
Where: \(A_s\) = Surface area of the solid sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(r\) = Radius of the sphere
Mass of Solid Sphere The mass (\(m_s\)) of a solid sphere can be calculated using the formula: \[ m_s = \text{Density} \times V_s \] Where: \(m_s\) = Mass of the solid sphere, \(\text{Density}\) = Density of the material making up the sphere (usually in \(kg/m^3\)), \(V_s\) = Volume of the solid sphere (calculated using the previous formula)
Introduction – A hollow sphere is also a three-dimensional geometric figure, but unlike a solid sphere, it has an empty space inside. – It consists of an outer shell or surface with a certain thickness and an inner empty region.
Characteristics – The hollow sphere has a well-defined outer radius (\(R\)), inner radius (\(r\)), volume, surface area, and mass. – It is partially filled with matter, mainly in the form of the outer shell.
Volume of Hollow Sphere The formula to calculate the volume (\(V_h\)) of a hollow sphere is given by: \[ V_h = \frac{4}{3} \pi (R^3 – r^3) \] Where: – \(V_h\) = Volume of the hollow sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(R\) = Outer radius of the sphere, \(r\) = Inner radius of the sphere
Surface Area of Hollow Sphere The formula to calculate the surface area (\(A_h\)) of a hollow sphere is given by: \[ A_h = 4 \pi (R^2 – r^2) \] Where: \(A_h\) = Surface area of the hollow sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(R\) = Outer radius of the sphere, \(r\) = Inner radius of the sphere
Mass of Hollow Sphere The mass (\(m_h\)) of a hollow sphere can be calculated using the formula: \[ m_h = \text{Density} \times V_h \] Where: \(m_h\) = Mass of the hollow sphere, \(\text{Density}\) = Density of the material making up the sphere (usually in \(kg/m^3\)), \(V_h\) = Volume of the hollow sphere (calculated using the previous formula)
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