Emeka forms a ball of clay with a radius of 3 centimeters (cm). He then reforms the clay into a cylinder of radius 2cm. What is the height of the cylinder in centimeters, rounded to the nearest tenth?

Calculate Volume of Sphere: First, calculate the volume of the ball of clay using the formula for the volume of a sphere, which is V = (4/3)πr^3, where r is the radius of the sphere. Given radius of the sphere (ball of clay) r = 3 cm, the volume V = (4/3)π(3 cm)^3.Calculate Volume of Cylinder: Perform the calculation for the volume of the sphere. V = (4/3)π(3 cm)^3 = (4/3)π(27 cm^3) = 36π cm^3.Solve for Cylinder Height: Since the clay is reformed into a cylinder without any loss of material, the volume of the cylinder will be equal to the volume of the sphere. Let's denote the height of the cylinder as h. The formula for the volume of a cylinder is V = πr^2h, where r is the radius of the cylinder. Given radius of the cylinder r = 2 cm, and the volume V = 36π cm^3, we have 36π cm^3 = π(2 cm)^2h.Simplify Equation for Height: Solve for the height h of the cylinder. 36π cm^3 = π(2 cm)^2h 36π cm^3 = π(4 cm^2)h To find h, divide both sides by π(4 cm^2). h = (36π cm^3) / (π(4 cm^2))Round Height to Nearest Tenth: Simplify the equation to find the height h. h = (36π cm^3) / (π(4 cm^2)) h = (36/4) cm h = 9 cmRound the height to the nearest tenth. The height h is already an integer value, so rounding to the nearest tenth will not change the value. h = 9.0 cm

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Emeka forms a ball of clay with a radius of 3 centimeters
(cm). He then reforms the clay into a cylinder of radius
2cm. What is the height of the cylinder in centimeters, rounded to the nearest tenth?

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

View step-by-step help

Full solution

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

3

centimeters

(\mathrm{cm})

. He then reforms the clay into a cylinder of radius

2 \mathrm{~cm}

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

Calculate Volume of Sphere: First, calculate the volume of the ball of clay using the formula for the volume of a sphere, which is $V = \frac{4}{3}\pi r^3$ , where $r$ is the radius of the sphere. $\newline$ Given radius of the sphere (ball of clay) $r = 3$ cm, the volume $V = \frac{4}{3}\pi(3\,\text{cm})^3$ . $\newline$ Calculation: $V = \left(\frac{4}{3}\right)\pi(3 \, \text{cm})^3 = \left(\frac{4}{3}\right)\pi(27 \, \text{cm}^3) = 36\pi \, \text{cm}^3$
Calculate Volume of Cylinder: Since the clay is reformed into a cylinder without any loss of material, the volume of the cylinder will be equal to the volume of the sphere. $\newline$ Let's denote the height of the cylinder as $h$ . The formula for the volume of a cylinder is $V = \pi r^2 h$ , where $r$ is the radius of the cylinder. $\newline$ Given radius of the cylinder $r = 2\,\text{cm}$ , and the volume $V = 36\pi\,\text{cm}^3$ , we have $36\pi\,\text{cm}^3 = \pi(2\,\text{cm})^2h$ .
Solve for Cylinder Height: Solve for the height $h$ of the cylinder. $\newline$ $36\pi \, \text{cm}^3 = \pi(2 \, \text{cm})^2h$ $\newline$ $36\pi \, \text{cm}^3 = \pi(4 \, \text{cm}^2)h$ $\newline$ To find $h$ , divide both sides by $\pi(4 \, \text{cm}^2)$ . $\newline$ $h = \frac{36\pi \, \text{cm}^3}{\pi(4 \, \text{cm}^2)}$
Simplify Equation for Height: Simplify the equation to find the height $h$ . $h = \frac{36\pi \, \text{cm}^3}{\pi(4 \, \text{cm}^2)}$ $h = \left(\frac{36}{4}\right) \, \text{cm}$ $h = 9 \, \text{cm}$ Round the height to the nearest tenth. $\newline$ The height $h$ is already an integer value, so rounding to the nearest tenth will not change the value. $h = 9.0 \, \text{cm}$