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An inflatable ball can be modeled as a hollow sphere. Jace measures the outer diameter of the ball to be 50 cm. If the material that makes up the ball has a thickness of 1 cm, find the total volume of material that makes up the ball. Round your answer to the nearest hundredth if necessary. (Note: diagram is not drawn to scale.)

An inflatable ball can be modeled as a hollow sphere. Jace measures the outer diameter-example-1
User Jan Giacomelli
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2 Answers

6 votes
6 votes

We can see that the total volume of material that makes up the ball is approximately 7,544.75 cm³.

To find the total volume of material that makes up the ball, we need to subtract the volume of the inner hollow part from the volume of the outer sphere.

The outer diameter of the ball is given as 50 cm. Since the diameter is twice the radius, the radius of the outer sphere is 50 cm / 2 = 25 cm.

The formula for the volume of a sphere is V = (4/3) × π × r³, where r is the radius.

Substituting the value of the radius, we get V_outer = (4/3) × π × (25 cm)³.

The thickness of the material that makes up the ball is given as 1 cm. Therefore, the inner radius is the outer radius minus the thickness: r_inner = 25 cm - 1 cm = 24 cm.

Using the same formula, we can calculate the volume of the inner hollow part: V_inner = (4/3) × π × (24 cm)³.

To find the total volume of material, we subtract the volume of the inner hollow part from the volume of the outer sphere: V_material = V_outer - V_inner.

Now, let's calculate the values:

V_outer = (4/3) × π × (25 cm)³ ≈ 65,449.85 cm³

V_inner = (4/3) × π × (24 cm)³ ≈ 57,905.09 cm³

Finally, we subtract V_inner from V_outer to find the total volume of material:

V_material ≈ V_outer - V_inner ≈ 65,449.85 cm³ - 57,905.09 cm³ ≈ 7,544.75 cm³. (rounded to the nearest hundredth)

Therefore, the total volume of material that makes up the ball is approximately 7,544.75 cm³.

User Slack Flag
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3.5k points
14 votes
14 votes

we have that

The volume of a sphere is given by the formula


V=(4)/(3)*pi*r^3

The interior diameter is equal to

D=50-2(1)=48 cm

r=48/2=24 cm

The volume is equal to the volume of the sphere with the outer diameter minus the volume of the sphere with the interior diameter

so


\begin{gathered} V=(4)/(3)*pi*[25^3-24^3] \\ V=7,544.01\text{ cm}^3 \end{gathered}

The volume is 7,544.01 cm3

User CppChase
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