The volume of a cylinder is given by [tex]$v=\pi r^2 h$[/tex], where [tex]$r$[/tex] is the radius of the cylinder and [tex]$h$[/tex] is the height. Which expression represents the volume of this can?
[tex]$3 \pi x^2+4 \pi x+16 \pi$[/tex]
[tex]$3 \pi x^2+16 \pi$[/tex]
[tex]$3 \pi x^3+32 \pi$[/tex]
[tex]$3 \pi x^3+20 \pi x^2+44 \pi x+32 \pi$[/tex]
Answer (1)
The volume of a cylinder is given by V = π r 2 h .
Analyze the possible relationships between the radius r , height h , and the variable x .
Consider that if both r and h are linear functions of x , the volume is a cubic polynomial in x .
Based on the analysis, the most plausible answer is a cubic polynomial: 3 π x 3 + 20 π x 2 + 44 π x + 32 π .
Explanation
Problem Analysis The volume of a cylinder is given by the formula V = π r 2 h , where r is the radius and h is the height. We are asked to identify which of the given expressions could represent the volume of a cylinder, given that the expressions are in terms of a variable x . Without knowing the specific relationships between r , h , and x , we cannot definitively determine the correct expression. However, we can analyze the structure of the volume formula and the given options to make an educated guess.
Analyzing the Options The volume formula V = π r 2 h tells us that the volume is proportional to the square of the radius and the height. If we assume that both r and h are linear functions of x , i.e., r = a x + b and h = c x + d for some constants a , b , c , d , then the volume would be:
V = π ( a x + b ) 2 ( c x + d ) = π ( a 2 x 2 + 2 ab x + b 2 ) ( c x + d ) = π ( a 2 c x 3 + ( a 2 d + 2 ab c ) x 2 + ( 2 ab d + b 2 c ) x + b 2 d ) .
This would result in a cubic polynomial in x .
If r is constant and h is a quadratic function of x , i.e., h = a x 2 + b x + c , then the volume would be:
V = π r 2 ( a x 2 + b x + c ) = π ( a r 2 x 2 + b r 2 x + c r 2 ) .
This would result in a quadratic polynomial in x .
Now let's examine the given options:
3 π x 2 + 4 π x + 16 π : This is a quadratic polynomial in x .
3 π x 2 + 16 π : This is also a quadratic polynomial in x .
3 π x 3 + 32 π : This is a cubic polynomial in x .
3 π x 3 + 20 π x 2 + 44 π x + 32 π : This is a cubic polynomial in x .
Making Assumptions Based on the analysis above, if both the radius and height are linear functions of x , the volume will be a cubic polynomial in x . Among the given options, there are two cubic polynomials. Without more information, it's difficult to choose between them. However, if we assume that the radius is constant and the height is a quadratic function of x , the volume will be a quadratic polynomial in x . Among the given options, there are two quadratic polynomials. Without more information, it's difficult to choose between them.
However, if we assume that the radius is proportional to x and the height is constant, i.e., r = a x and h = b , then the volume would be:
V = π ( a x ) 2 b = π a 2 x 2 b = π a 2 b x 2 = k x 2 , where k = π a 2 b .
In this case, the volume is a quadratic polynomial in x .
If we assume that the radius is constant and the height is proportional to x , i.e., r = a and h = b x , then the volume would be:
V = π a 2 ( b x ) = π a 2 b x = k x , where k = π a 2 b .
In this case, the volume is a linear polynomial in x .
Conclusion Given the options, the most likely scenario is that both the radius and height are linear functions of x , which would result in a cubic polynomial. Therefore, we should choose one of the cubic polynomials.
Without additional information, we cannot definitively determine the correct expression. However, based on the possible relationships between r , h , and x , the most plausible answer is a cubic polynomial in x .
Examples
Understanding the volume of cylinders is crucial in various real-world applications. For example, when designing storage tanks for liquids or gases, engineers use the formula V = π r 2 h to calculate the required dimensions to hold a specific volume. Similarly, in packaging design, knowing the volume helps determine the amount of material needed to create cylindrical containers for products like canned goods or beverages. This ensures efficient use of resources and minimizes waste.
