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A soap company changes the design of its soap from a cone to a sphere. The cone had a height of 3 centimeters
(cm) and a radius of
2cm. The sphere has a diameter of
3cm. The new design contains
(pi )/(f) cubic centimeters more soap than the old design. What is the value of f?

A soap company changes the design of its soap from a cone to a sphere. The cone had a height of $3$ centimeters $(\mathrm{cm})$ and a radius of $2 \mathrm{~cm}$ . The sphere has a diameter of $3 \mathrm{~cm}$ . The new design contains $\frac{\pi}{f}$ cubic centimeters more soap than the old design. What is the value of $f$ ?

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Scale drawings: word problems

Full solution

Q. A soap company changes the design of its soap from a cone to a sphere. The cone had a height of

3

centimeters

(\mathrm{cm})

and a radius of

2 \mathrm{~cm}

. The sphere has a diameter of

3 \mathrm{~cm}

. The new design contains

\frac{\pi}{f}

cubic centimeters more soap than the old design. What is the value of

f

?

Calculate Cone Volume: First, we need to calculate the volume of the original cone-shaped soap. $\newline$ The formula for the volume of a cone is $V = \frac{1}{3}\pi r^2 h$ , where $r$ is the radius and $h$ is the height. $\newline$ Given: radius $r = 2$ cm, height $h = 3$ cm. $\newline$ Let's calculate the volume of the cone. $\newline$ $V_{\text{cone}} = \frac{1}{3}\pi(2\,\text{cm})^2(3\,\text{cm}) = \frac{1}{3}\pi(4\,\text{cm}^2)(3\,\text{cm}) = 4\pi\,\text{cm}^3$ .
Calculate Sphere Volume: Next, we calculate the volume of the new sphere-shaped soap. $\newline$ The formula for the volume of a sphere is $V = \frac{4}{3}\pi r^3$ , where $r$ is the radius. $\newline$ Given: diameter of the sphere = $3$ cm, so the radius $r = \frac{\text{diameter}}{2} = \frac{3\,\text{cm}}{2} = 1.5$ cm. $\newline$ Let's calculate the volume of the sphere. $\newline$ $V_{\text{sphere}} = \frac{4}{3}\pi(1.5\,\text{cm})^3 = \frac{4}{3}\pi(3.375\,\text{cm}^3) = 4.5\pi\,\text{cm}^3$ .
Find Volume Difference: Now, we find the difference in volume between the sphere and the cone to determine how much more soap the new design contains. $\newline$ Difference in volume = $V_{\text{sphere}} - V_{\text{cone}} = 4.5\pi \, \text{cm}^3 - 4\pi \, \text{cm}^3 = 0.5\pi \, \text{cm}^3$ .
Set Up Equation: The problem states that the new design contains $\frac{\pi}{f}$ cubic centimeters more soap than the old design. $\newline$ We have found that the difference in volume is $0.5\pi$ cm³. $\newline$ So, we set up the equation $\frac{\pi}{f} = 0.5\pi$ .
Solve for f: To find the value of f, we solve the equation $\frac{\pi}{f} = 0.5\pi$ . We can divide both sides of the equation by $\pi$ to get $\frac{1}{f} = 0.5$ . Finally, we solve for f by taking the reciprocal of $0.5$ . $\newline$ $f = \frac{1}{0.5} = 2$ .

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