Which is the correct first step in finding the area of the base of a cylinder with a volume of [tex]$140 \pi$[/tex] cubic meters and a height of 12 meters?
A. [tex]$V=B h$[/tex]
[tex]$12=B(140 \pi)$[/tex]
B. [tex]$V=B h$[/tex]
[tex]$V=140 \pi+(12)$[/tex]
C. [tex]$V=B h$[/tex]
[tex]$V=140 \pi(12)$[/tex]
D. [tex]$V=B h$[/tex]
[tex]$140 \pi=B(12)$[/tex]
Answer (2)
Substitute the given volume V = 140 π and height h = 12 into the cylinder volume formula V = B h .
Obtain the equation 140 π = B ( 12 ) .
Compare the obtained equation with the given options.
Identify the correct first step: 140 π = B ( 12 ) .
Explanation
Problem Analysis and Setup We are given a cylinder with a volume of 140 π cubic meters and a height of 12 meters. We want to find the area of the base of the cylinder. The formula for the volume of a cylinder is given by V = B h , where V is the volume, B is the area of the base, and h is the height. Our goal is to identify the correct first step in solving for B .
Substitution of Values We are given that the volume V = 140 π cubic meters and the height h = 12 meters. We substitute these values into the formula V = B h to get: 140 π = B ( 12 ) This equation represents the correct substitution of the given values into the volume formula.
Comparison with Given Options Now, let's compare this equation with the given options:
12 = B ( 140 π ) : This is incorrect because it swaps the volume and height values.
V = 140 π + ( 12 ) : This is incorrect because it adds the volume and height, instead of using the volume formula.
V = 140 π ( 12 ) : This is incorrect because it multiplies the volume by the height, which is not part of the volume formula.
140 π = B ( 12 ) : This is the correct equation, as it accurately substitutes the given volume and height into the formula V = B h .
Conclusion Therefore, the correct first step is to substitute the given values into the formula V = B h , which results in the equation: 140 π = B ( 12 ) This equation correctly represents the relationship between the volume, base area, and height of the cylinder.
Examples
Understanding the volume of cylinders is crucial in many real-world applications. For example, when designing cylindrical storage tanks for liquids or gases, engineers need to calculate the base area to ensure the tank can hold the required volume. If a tank needs to hold 140 π cubic meters and has a height of 12 meters, finding the base area using the formula V = B h is the first step in determining the tank's dimensions. This ensures the tank meets the storage requirements and is structurally sound. Another example is calculating the volume of a can of food or a pipe.
The correct first step in finding the area of the base of the cylinder is to use the volume formula V = B h and substitute the given values, which results in the equation 140 π = B ( 12 ) . This corresponds to option D. Subsequently, we can solve for B from this equation.
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