A credit card-sized piece is needed for the base of a volcano model. The volcano is 46 centimeters tall and has a volume of 800 cubic centimeters. Which equation can be used to find the area of the circular base?
[tex]$45=\frac{1}{3}(\xi)^2(800)$[/tex]
[tex]$45=\frac{1}{3}(5)(300)$[/tex]
[tex]$300=\frac{7}{3}\left(8^2\right)(45)$[/tex]
[tex]$300=\frac{9}{3}(B)(-45)$[/tex]
Answer (2)
The problem provides the height and volume of a cone-shaped volcano model and asks for the equation to find the base area.
Recall the formula for the volume of a cone: V = 3 1 B h .
Substitute the given values, V = 800 and h = 46 , into the formula: 800 = 3 1 B ( 46 ) .
The equation to find the area of the circular base is 800 = 3 1 B ( 46 ) .
Explanation
Problem Analysis and Volume Formula Let's analyze the problem. We are given the height and volume of a volcano model, which we can assume to be cone-shaped. We need to find an equation that relates these quantities to the area of the circular base. The formula for the volume of a cone is:
V = 3 1 B h
where V is the volume, B is the area of the base, and h is the height.
Substitute Given Values We are given that the height h = 46 cm and the volume V = 800 cubic cm. We can substitute these values into the formula:
800 = 3 1 B ( 46 )
Solve for B Now, let's compare this equation with the given options to see which one matches. None of the provided options exactly match the derived equation. However, we can manipulate our equation to isolate B :
800 = 3 46 B
B = 46 3 × 800 = 46 2400 ≈ 52.17
Final Equation The equation that can be used to find the area of the circular base is:
800 = 3 1 B ( 46 )
Examples
Understanding the relationship between volume, base area, and height is crucial in various real-world applications. For instance, when designing containers or structures, engineers need to calculate the required dimensions to achieve a specific volume. Similarly, in architecture, knowing the base area and height helps in estimating the volume of materials needed for construction. This principle also applies in fields like fluid dynamics, where calculating the volume of a liquid in a container is essential for various processes.
To find the area of the base of the volcano modeled as a cone, we use the volume formula V = 3 1 B h . Given that the volume is 800 cubic cm and the height is 46 cm, we derive the equation 800 = 3 1 B ( 46 ) . This equation can be rearranged to find the area of the base, B .
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