
Two similar cones have volumes cu. units and cu. units. If the curved surface area of the smaller cone is sq. units, then what is the curved surface area of the larger cone?
Answer
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Hint: Here, we need to find the curved surface area of the larger cone. We will use the fact that the ratio of the heights and radii of two similar cones is equal. We will write the radius and height of one cone in terms of the radius and height of the other cone. Then, we will use the given information, formula for volume, formula for slant heights, and finally, the formula for the curved surface area of a cone to find the required area.
Formula used:
We will use the following formulas:
1.The volume of a cone is given by the formula , where is the radius of the base of the cone, and is the height of the cone.
2.The slant height of a cone is given by the formula , where is the radius of the base of the cone, and is the height of the cone.
3.The curved surface area of a cone is given by the formula , where is the radius of the base of the cone, and is the slant height of the cone.
Complete step-by-step answer:
Let the height of the smaller and larger cone be and units respectively.
Let the radius of the smaller and larger cone be and units respectively.
So, now we will draw the diagrams of the cone.
It is given that the two cones are similar. This means that the ratios of the dimensions of the cones are equal.
Thus, the ratio of the heights of the cones is equal to the ratio of the radii of the cone.
Therefore, we get
Let .
Therefore, we get the equations
and
Now, we will calculate the volumes of the two cones.
Substituting and in the formula for volume of a cone , we get
Volume of the smaller cone
Substituting and in the equation, we get
Volume of the smaller cone
Simplifying the expression, we get
Volume of the smaller cone
Substituting and in the formula for volume of a cone , we get
Volume of the larger cone
It is given that the volume of the smaller and larger cone is cu. units and cu. units respectively.
Thus, we get the equations
and
We will divide the two equations for volumes to form an equation in terms of .
Dividing the equation by the equation , we get
Simplifying the expression, we get
Taking the cube root of both sides of the equation, we get
Substituting in the equation , we get
Substituting in the equation , we get
Now, we will calculate the slant heights of the two cones.
Let the slant height of the smaller and larger cone be and units respectively.
The slant height of a cone is given by the formula , where is the radius of the base of the cone, and is the height of the cone.
Substituting , and in the formula for slant height, we get
Substituting , and in the formula for slant height, we get
Now, we will calculate the curved surface areas of the two cones.
The curved surface area of a cone is given by the formula , where is the radius of the base of the cone, and is the slant height of the cone.
Substituting and in the formula for curved surface area of a cone, we get
Curved surface area of smaller cone
Substituting in the equation, we get
Curved surface area of smaller cone
It is given that the curved surface area of the smaller cone is sq. units.
Thus, we get the equation
Substituting and in the equation, we get
Simplifying the expression, we get
Thus, we get
Multiplying both sides of the equation by 4, we get
Substituting and in the formula for curved surface area of a cone, we get
Curved surface area of larger cone
Substituting in the equation, we get
Curved surface area of larger cone
Substituting in the equation, we get
Curved surface area of larger cone
Therefore, we get the curved surface area of the larger cone as sq. units.
Note: We have not substituted because the volumes of the two cones, and the curved surface area of the smaller cone is given in the question in terms of . Therefore, substituting the value and multiplying the expression is not needed.
We calculated the curved surface area of the cone in the question. The curved surface area of a cone is the area of the rounded surface of a cone. It is equal to the difference in the total surface area of the cone, and the area of the circular base of the cone.
Formula used:
We will use the following formulas:
1.The volume of a cone is given by the formula
2.The slant height of a cone is given by the formula
3.The curved surface area of a cone is given by the formula
Complete step-by-step answer:
Let the height of the smaller and larger cone be
Let the radius of the smaller and larger cone be
So, now we will draw the diagrams of the cone.

It is given that the two cones are similar. This means that the ratios of the dimensions of the cones are equal.
Thus, the ratio of the heights of the cones is equal to the ratio of the radii of the cone.
Therefore, we get
Let
Therefore, we get the equations
Now, we will calculate the volumes of the two cones.
Substituting
Volume of the smaller cone
Substituting
Simplifying the expression, we get
Substituting
Volume of the larger cone
It is given that the volume of the smaller and larger cone is
Thus, we get the equations
and
We will divide the two equations for volumes to form an equation in terms of
Dividing the equation
Simplifying the expression, we get
Taking the cube root of both sides of the equation, we get
Substituting
Substituting
Now, we will calculate the slant heights of the two cones.
Let the slant height of the smaller and larger cone be
The slant height of a cone is given by the formula
Substituting
Substituting
Now, we will calculate the curved surface areas of the two cones.
The curved surface area of a cone is given by the formula
Substituting
Curved surface area of smaller cone
Substituting
Curved surface area of smaller cone
It is given that the curved surface area of the smaller cone is
Thus, we get the equation
Substituting
Simplifying the expression, we get
Thus, we get
Multiplying both sides of the equation by 4, we get
Substituting
Curved surface area of larger cone
Substituting
Curved surface area of larger cone
Substituting
Curved surface area of larger cone
Therefore, we get the curved surface area of the larger cone as
Note: We have not substituted
We calculated the curved surface area of the cone in the question. The curved surface area of a cone is the area of the rounded surface of a cone. It is equal to the difference in the total surface area of the cone, and the area of the circular base of the cone.
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