Determine The Volume Of The Cone If It Were Dilated By A Scale Factor Of 1/3. Calculate Your Answer In Terms Of Pie, And Round Your Final Answer To The Nearest Hundredth. The Cone Has A Radius Of 10 And A Height Of 25.
In geometry, dilation is a transformation that changes the size of a figure. When a figure is dilated by a scale factor, its size is multiplied by that factor. In this article, we will explore how to determine the volume of a cone if it were dilated by a scale factor of 1/3.
What is a Cone?
A cone is a three-dimensional shape that tapers smoothly from a circular base to a point called the apex. The radius of the base is the distance from the center of the base to the edge, while the height is the distance from the base to the apex.
Properties of a Cone
The volume of a cone is given by the formula:
V = (1/3)πr²h
where V is the volume, π is a mathematical constant approximately equal to 3.14, r is the radius of the base, and h is the height of the cone.
Dilating a Cone
When a cone is dilated by a scale factor of 1/3, its size is reduced by a factor of 1/3. This means that the radius of the base is reduced to 1/3 of its original value, and the height is also reduced to 1/3 of its original value.
Calculating the New Volume
To calculate the new volume of the dilated cone, we need to substitute the new values of the radius and height into the formula for the volume of a cone.
Let's assume the original radius of the cone is 10 and the original height is 25. When the cone is dilated by a scale factor of 1/3, the new radius is 10/3 and the new height is 25/3.
The new volume of the dilated cone is:
V = (1/3)π(10/3)²(25/3)
Simplifying the Expression
To simplify the expression, we can start by squaring the new radius:
(10/3)² = 100/9
Now, substitute this value back into the expression for the new volume:
V = (1/3)π(100/9)(25/3)
Multiplying the Fractions
To multiply the fractions, we can multiply the numerators and the denominators separately:
V = (1/3)π(100 × 25) / (9 × 3)
V = (1/3)π(2500) / 27
Simplifying the Expression Further
Now, we can simplify the expression further by multiplying the numerator and the denominator by 3:
V = π(2500) / 81
Rounding the Final Answer
Finally, we can round the final answer to the nearest hundredth. To do this, we need to calculate the value of π(2500) / 81 and round it to two decimal places.
π(2500) ≈ 7853.98
Now, divide this value by 81:
7853.98 / 81 ≈ 97.55
Therefore, the volume of the dilated cone is approximately 97.55 cubic units.
Conclusion
In the previous article, we explored how to determine the volume of a cone if it were dilated by a scale factor of 1/3. However, we may still have some questions about the concept of dilation and how it applies to cones. In this article, we will answer some frequently asked questions about dilating a cone.
Q: What is dilation in geometry?
A: Dilation is a transformation that changes the size of a figure. When a figure is dilated by a scale factor, its size is multiplied by that factor.
Q: What is a scale factor?
A: A scale factor is a number that is used to multiply the size of a figure. In the case of dilation, the scale factor is used to multiply the dimensions of the figure.
Q: How does dilation affect the volume of a cone?
A: When a cone is dilated by a scale factor, its volume is also multiplied by that factor. This means that the volume of the dilated cone is equal to the original volume multiplied by the cube of the scale factor.
Q: What is the formula for the volume of a dilated cone?
A: The formula for the volume of a dilated cone is:
V = (1/3)π(r/s)²(h/s)³
where V is the volume, π is a mathematical constant approximately equal to 3.14, r is the original radius, s is the scale factor, and h is the original height.
Q: How do I calculate the new volume of a dilated cone?
A: To calculate the new volume of a dilated cone, you need to substitute the new values of the radius and height into the formula for the volume of a cone. You also need to multiply the original volume by the cube of the scale factor.
Q: What is the difference between dilation and scaling?
A: Dilation and scaling are both transformations that change the size of a figure. However, dilation is a more general term that refers to any transformation that changes the size of a figure, while scaling specifically refers to a transformation that changes the size of a figure by a fixed factor.
Q: Can I dilate a cone by a scale factor greater than 1?
A: Yes, you can dilate a cone by a scale factor greater than 1. However, this will result in a larger cone with a greater volume.
Q: Can I dilate a cone by a scale factor less than 1?
A: Yes, you can dilate a cone by a scale factor less than 1. However, this will result in a smaller cone with a smaller volume.
Q: What is the effect of dilation on the surface area of a cone?
A: When a cone is dilated by a scale factor, its surface area is also multiplied by that factor. This means that the surface area of the dilated cone is equal to the original surface area multiplied by the square of the scale factor.
Conclusion
In this article, we answered some frequently asked questions about dilating a cone. We covered topics such as dilation, scale factors, and the effect of dilation on the volume and surface area of a cone. We hope that this article has been helpful in clarifying any questions you may have had about dilating a cone.