Question - Calculating Cone Radius With Given Volume and Height
Solution:
To find the radius of the cone, we can use the formula for the volume of a cone:\[ V = \frac{1}{3}\pi r^2 h \]Where:- $$ V $$ is the volume of the cone,- $$ r $$ is the radius of the base of the cone,- $$ h $$ is the height of the cone,- $$ \pi $$ is approximately 3.14 (given).The volume ($$ V $$) is given as 37.68 cubic millimeters, and the height ($$ h $$) is given as 4 millimeters. To find $$ r $$, we can rearrange the formula to solve for $$ r^2 $$:\[ r^2 = \frac{3V}{\pi h} \]Now we'll use the given values:\[ r^2 = \frac{3 \times 37.68}{3.14 \times 4} \]First, let's calculate the numerator of the fraction:\[ 3 \times 37.68 = 113.04 \]And the denominator of the fraction:\[ 3.14 \times 4 = 12.56 \]Next, we'll divide these two results to find $$ r^2 $$:\[ r^2 = \frac{113.04}{12.56} \approx 9 \]Now take the square root of 9 to find the radius $$ r $$:\[ r = \sqrt{9} = 3 \]So the radius $$ r $$ of the cone is 3 millimeters. Since the problem asks to round our answer to the nearest hundredth, our final answer in two decimal places would still be:\[ r \approx 3.00 \text{ millimeters} \]
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