Emeka forms a ball of clay with a radius of $3\,\text{centimeters}$ . He then reforms the clay into a cylinder of radius $2$ . What is the height of the cylinder in centimeters, rounded to the nearest tenth?

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Q. Emeka forms a ball of clay with a radius of

3\,\text{centimeters}

. He then reforms the clay into a cylinder of radius

2

. What is the height of the cylinder in centimeters, rounded to the nearest tenth?

Find Volume of Ball: We need to find the volume of the ball of clay first, since the volume of the clay remains the same when it is reshaped into a cylinder. The formula for the volume of a sphere (ball) is $V = \frac{4}{3}\pi r^3$ , where $r$ is the radius of the sphere.
Calculate Volume of Ball: Let's calculate the volume of the ball of clay with a radius of $3$ centimeters. Using the formula $V = \frac{4}{3}\pi r^3$ and $\pi \approx 3.14$ , we get $V = \frac{4}{3} \times 3.14 \times 3^3$ .
Find Volume of Cylinder: Now, we perform the calculation: $V = \frac{4}{3} \times 3.14 \times 27$ . First, calculate $3^3 = 27$ , then multiply by $3.14$ to get $84.78$ , and finally multiply by $\frac{4}{3}$ to find the volume.
Calculate Volume of Cylinder: The volume of the ball of clay is $V = \frac{4}{3} \times 84.78$ , which equals $113.04$ cubic centimeters after rounding to two decimal places.
Find Height of Cylinder: Next, we need to find the volume of the cylinder using the formula $V = \pi r^2 h$ , where $r$ is the radius and $h$ is the height of the cylinder. Since the volume of the clay doesn't change, the volume of the cylinder will be the same as the volume of the ball, which is $113.04$ cubic centimeters.
Find Height of Cylinder: Next, we need to find the volume of the cylinder using the formula $V = \pi r^2 h$ , where $r$ is the radius and $h$ is the height of the cylinder. Since the volume of the clay doesn't change, the volume of the cylinder will be the same as the volume of the ball, which is $113.04$ cubic centimeters.We know the volume $V$ and the radius $r$ of the cylinder, so we can rearrange the formula to solve for the height $h$ : $h = \frac{V}{\pi r^2}$ .
Find Height of Cylinder: Next, we need to find the volume of the cylinder using the formula $V = \pi r^2 h$ , where $r$ is the radius and $h$ is the height of the cylinder. Since the volume of the clay doesn't change, the volume of the cylinder will be the same as the volume of the ball, which is $113.04$ cubic centimeters.We know the volume $V$ and the radius $r$ of the cylinder, so we can rearrange the formula to solve for the height $h$ : $h = \frac{V}{\pi r^2}$ .Plugging in the values, we get $h = \frac{113.04}{3.14 \times 2^2}$ . First, calculate $2^2 = 4$ , then multiply by $r$ $0$ to get $r$ $1$ , and finally divide $113.04$ by $r$ $1$ to find the height.
Find Height of Cylinder: Next, we need to find the volume of the cylinder using the formula $V = \pi r^2 h$ , where $r$ is the radius and $h$ is the height of the cylinder. Since the volume of the clay doesn't change, the volume of the cylinder will be the same as the volume of the ball, which is $113.04$ cubic centimeters.We know the volume $V$ and the radius $r$ of the cylinder, so we can rearrange the formula to solve for the height $h$ : $h = \frac{V}{\pi r^2}$ .Plugging in the values, we get $h = \frac{113.04}{3.14 \times 2^2}$ . First, calculate $2^2 = 4$ , then multiply by $r$ $0$ to get $r$ $1$ , and finally divide $113.04$ by $r$ $1$ to find the height.The height of the cylinder is $r$ $4$ , which equals approximately $r$ $5$ centimeters when rounded to the nearest tenth.