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 fπ cubic centimeters more soap than the old design. What is the value of f ?
Q. 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 fπ 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.The formula for the volume of a cone is V=31πr2h, where r is the radius and h is the height.Given: radius r=2 cm, height h=3 cm.Let's calculate the volume of the cone.Vcone=31π(2cm)2(3cm)=31π(4cm2)(3cm)=4πcm3.
Calculate Sphere Volume: Next, we calculate the volume of the new sphere-shaped soap.The formula for the volume of a sphere is V=34πr3, where r is the radius.Given: diameter of the sphere = 3 cm, so the radius r=2diameter=23cm=1.5 cm.Let's calculate the volume of the sphere.Vsphere=34π(1.5cm)3=34π(3.375cm3)=4.5πcm3.
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.Difference in volume = Vsphere−Vcone=4.5πcm3−4πcm3=0.5πcm3.
Set Up Equation: The problem states that the new design contains fπ cubic centimeters more soap than the old design.We have found that the difference in volume is 0.5π cm³.So, we set up the equation fπ=0.5π.
Solve for f: To find the value of f, we solve the equation fπ=0.5π. We can divide both sides of the equation by π to get f1=0.5. Finally, we solve for f by taking the reciprocal of 0.5. f=0.51=2.