In an effort to make their product seem like a better bargain, the Chock-a-Lot candy
company increased the size of their chocolate balls, from a 2-cm diameter to a 3-cm diameter,
without increasing the price. In fact, the new balls still contain the same amount of chocolate,
because they are hollow spherical shells, while the 2-cm balls are solid chocolate. How thick
are the spherical chocolate shells that Chock-a-Lot is now selling?
To calculate the thickness of the spherical chocolate shell, we need to find the difference in volume between the 3-cm diameter hollow shell and the 2-cm diameter solid chocolate ball.
The formula for the volume of a sphere is V = (4/3)πr^3, where r is the radius.
For the 3-cm diameter hollow shell, the radius is 3/2 = 1.5 cm.
The volume of the hollow shell is V1 = (4/3)π(1.5)^3 = 14.13 cm^3.
For the 2-cm diameter solid chocolate ball, the radius is 2/2 = 1 cm.
The volume of the solid chocolate ball is V2 = (4/3)π(1)^3 = 4.19 cm^3.
The difference in volume between the hollow shell and the solid ball is V_diff = V1 - V2 = 14.13 cm^3 - 4.19 cm^3 = 9.94 cm^3.
Now, let's find the volume of the chocolate shell. Since the shell is hollow, its volume consists of the difference in volume we calculated above.
The volume of the chocolate shell is V_shell = V_diff = 9.94 cm^3.
Finally, let's calculate the thickness of the chocolate shell. The thickness can be determined by subtracting the volume of the inner cavity (which is a solid chocolate ball) from the volume of the shell and dividing by the surface area of the shell.
The surface area of a sphere is A = 4πr^2.
The surface area of the 3-cm diameter hollow shell is A1 = 4π(1.5)^2 = 28.27 cm^2.
To find the thickness, we divide the difference in volume by the surface area of the shell. The formula is:
Thickness = V_shell / A1 = 9.94 cm^3 / 28.27 cm^2 ≈ 0.351 cm.
Therefore, the spherical chocolate shells that Chock-a-Lot is selling have a thickness of approximately 0.351 cm.
To find the thickness of the spherical chocolate shells that Chock-a-Lot is now selling, we need to calculate the difference in volume between the new hollow shell and the original solid chocolate ball.
The volume of a hollow sphere can be calculated using the formula:
V = (4/3)π(R1^3 - R2^3)
Where:
V = Volume
π = Pi, approximately 3.14159
R1 = Radius of the outer shell (1.5 cm, half the diameter of the new 3-cm ball)
R2 = Radius of the inner solid chocolate ball (1 cm, half the diameter of the original 2-cm ball)
Plugging in the values, we get:
V = (4/3)π((1.5)^3 - (1)^3)
= (4/3)π(3.375 - 1)
= (4/3)π(2.375)
≈ (4/3)(3.14159)(2.375)
≈ 3.14159(7.91719)
≈ 24.78869 cm^3
Since the hollow shell contains the same amount of chocolate as the original solid ball, the volume remains the same.
Now, let's calculate the volume of the original solid chocolate ball using the formula for the volume of a sphere:
V = (4/3)π(R^3)
Where:
V = Volume (same as the hollow shell, 24.78869 cm^3)
π = Pi, approximately 3.14159
R = Radius of the original solid chocolate ball (1 cm)
Plugging in the values, we get:
24.78869 = (4/3)π(1)^3
= (4/3)π
≈ (4/3)(3.14159)
≈ 4.18879 cm^3
Now, we can calculate the thickness of the spherical chocolate shells by subtracting the volume of the inner solid ball from the volume of the outer shell:
Shell Thickness = Volume of Outer Shell - Volume of Inner Solid Ball
= 24.78869 cm^3 - 4.18879 cm^3
≈ 20.5999 cm^3
Therefore, the thickness of the spherical chocolate shells that Chock-a-Lot is selling is approximately 20.5999 cm^3.
To find the thickness of the spherical chocolate shells that Chock-a-Lot is currently selling, we can subtract the radius of the solid chocolate balls from the radius of the hollow chocolate shells.
1. Start by calculating the radius of the 2-cm solid chocolate balls:
- Diameter = 2 cm
- Radius = Diameter / 2 = 2 cm / 2 = 1 cm
2. Next, calculate the radius of the 3-cm hollow chocolate shells:
- Diameter = 3 cm
- Radius = Diameter / 2 = 3 cm / 2 = 1.5 cm
3. Finally, calculate the thickness by subtracting the radius of the solid chocolate balls from the radius of the hollow chocolate shells:
- Thickness = Radius of hollow shells - Radius of solid balls = 1.5 cm - 1 cm = 0.5 cm
Therefore, the spherical chocolate shells that Chock-a-Lot is currently selling have a thickness of 0.5 cm.