Which statements are true? Check all that apply.
The radius of the cone is 9 units.
The height of the cone is 15 units.
The height of the cone is 12 units.
The volume of the cone is represented by the expression [tex]$\frac{1}{3} \pi(15)^2(9)$[/tex].
The volume of the cone is represented by the expression [tex]$\frac{1}{3} \pi(9)^2(12)$[/tex].
Answer (1)
The volume of a cone is calculated using the formula V = 3 1 π r 2 h .
If the radius is 9 and the height is 12, the volume is V = 3 1 π ( 9 ) 2 ( 12 ) = 324 π .
If the radius is 15 and the height is 9, the volume is V = 3 1 π ( 15 ) 2 ( 9 ) = 675 π .
Therefore, the true statements are: The radius of the cone is 9 units, the height of the cone is 12 units, and the volume of the cone is represented by the expression 3 1 π ( 9 ) 2 ( 12 ) .
Explanation
Problem Analysis and Volume Formula Let's analyze the given statements about the cone's dimensions and volume to determine which ones are true. We'll use the formula for the volume of a cone, which is:
V = 3 1 π r 2 h
where V is the volume, r is the radius, and h is the height.
Volume Calculation with Radius 9 and Height 12 First, let's consider the case where the radius of the cone is 9 units and the height is 12 units. We can calculate the volume using the formula:
V = 3 1 π ( 9 ) 2 ( 12 )
V = 3 1 π ( 81 ) ( 12 )
V = 3 1 π ( 972 )
V = 324 π
The result of the operation is 324 π ≈ 1017.88 cubic units.
Volume Calculation with Radius 15 and Height 9 Now, let's consider the case where the radius is 15 units and the height is 9 units. The volume would be:
V = 3 1 π ( 15 ) 2 ( 9 )
V = 3 1 π ( 225 ) ( 9 )
V = 3 1 π ( 2025 )
V = 675 π
The result of the operation is 675 π ≈ 2120.58 cubic units.
Evaluating Each Statement Now, let's evaluate each statement:
The radius of the cone is 9 units. This statement could be true, as we considered this case. However, we don't know for sure if this is the only possibility.
The height of the cone is 15 units. This statement could be true if the radius is 9 units, but we don't have enough information to confirm.
The height of the cone is 12 units. This statement could be true if the radius is 9 units, but we don't have enough information to confirm.
The volume of the cone is represented by the expression 3 1 π ( 15 ) 2 ( 9 ) . This corresponds to the case where the radius is 15 and the height is 9, which we calculated to be 675 π .
The volume of the cone is represented by the expression 3 1 π ( 9 ) 2 ( 12 ) . This corresponds to the case where the radius is 9 and the height is 12, which we calculated to be 324 π .
Identifying True Statements Based on our calculations, the following statements are true:
The radius of the cone is 9 units, and the height of the cone is 12 units, then the volume of the cone is represented by the expression 3 1 π ( 9 ) 2 ( 12 ) .
Therefore, the true statements are:
The radius of the cone is 9 units.
The height of the cone is 12 units.
The volume of the cone is represented by the expression 3 1 π ( 9 ) 2 ( 12 ) .
Examples
Understanding the volume of cones is crucial in various real-world applications. For instance, architects use these calculations to design structures like conical roofs, ensuring they can withstand specific loads and environmental conditions. Similarly, engineers apply cone volume formulas to optimize the design of funnels and storage containers, maximizing efficiency and minimizing material usage. Even in culinary arts, chefs use the concept of cone volume to estimate the amount of ingredients needed for desserts like ice cream cones, ensuring consistent and visually appealing servings.
