A cone has a height that measures $(x+2)$ centimeters and a volume of 48 cubic centimeters. Which expression represents the radius of the cone?
$\sqrt{\frac{144}{\pi(x+2)}}$
$\frac{16}{\pi(x+2)}$
$\frac{144(x+2)}{\pi}$
$16 \pi(x+2)$
Answer (2)
Recall the formula for the volume of a cone: V = 3 1 π r 2 h .
Substitute the given values: 48 = 3 1 π r 2 ( x + 2 ) .
Solve for r 2 : r 2 = π ( x + 2 ) 144 .
Solve for r : r = π ( x + 2 ) 144 .
π ( x + 2 ) 144
Explanation
Problem Analysis We are given a cone with height h = ( x + 2 ) centimeters and volume V = 48 cubic centimeters. We need to find an expression for the radius r of the cone.
Recall the volume formula The formula for the volume of a cone is given by: V = 3 1 π r 2 h
Substitute the given values Substitute the given values into the formula: 48 = 3 1 π r 2 ( x + 2 ) Now, we need to solve for r .
Isolate r 2 Multiply both sides by 3: 144 = π r 2 ( x + 2 ) Divide both sides by π ( x + 2 ) :
r 2 = π ( x + 2 ) 144
Solve for r Take the square root of both sides to solve for r :
r = π ( x + 2 ) 144
Final Answer Therefore, the expression representing the radius of the cone is π ( x + 2 ) 144 .
Examples
Cones are commonly found in everyday life, from ice cream cones to traffic cones. Understanding how the volume, radius, and height of a cone are related allows us to solve practical problems. For example, if you're designing a paper cup in the shape of a cone and you know the desired volume and height, you can use the formula to determine the required radius. This ensures the cup holds the intended amount of liquid. The formula for the volume of a cone is V = 3 1 π r 2 h , where V is the volume, r is the radius, and h is the height. By rearranging this formula, we can solve for any of the variables if we know the other two.
To determine the radius of a cone with a height of ( x + 2 ) cm and volume of 48 cm³, we use the formula for the volume of the cone. By substituting the values and rearranging, we find that the expression for the radius is oxed{\sqrt{\frac{144}{\pi(x + 2)}}} . This matches the first option in the question prompt.
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