Elijah is planning a water day and wants to pack a sandwich that won’t get soggy. He wants to get a plastic container in the shape of a cylinder with two hemispheres attached on either end. The height of the cylinder is equal to its diameter. If the volume of the cylinder is 60 in.3

Question

, what is the volume of the entire plastic container? Round to the nearest tenths place if needed.

Answer 1

To find the volume of the entire plastic container, we need to calculate the volume of the cylindrical part and the volumes of the two hemispherical ends.

Step 1: Define the dimensions of the cylinder

Let the diameter of the cylinder be \( d \) and the radius \( r \) be half of the diameter: \[ r = \frac{d}{2} \]

Given that the height \( h \) of the cylinder is equal to its diameter, we have: \[ h = d \]

Step 2: Volume of the cylinder

The volume \( V_c \) of the cylinder can be calculated using the formula: \[ V_c = \pi r^2 h \]

Substituting \( h = d \) and \( r = \frac{d}{2} \) into the formula: \[ V_c = \pi \left(\frac{d}{2}\right)^2 d = \pi \left(\frac{d^2}{4}\right) d = \frac{\pi d^3}{4} \]

Step 3: Set up the equation with the known volume

We are given that the volume of the cylinder is \( 60 , \text{in}^3 \): \[ \frac{\pi d^3}{4} = 60 \]

Step 4: Solve for \( d^3 \)

Multiply both sides by 4: \[ \pi d^3 = 240 \] Now divide by \( \pi \): \[ d^3 = \frac{240}{\pi} \]

Step 5: Calculate \( d \)

To find \( d \), we take the cube root: \[ d = \sqrt[3]{\frac{240}{\pi}} \]

Approximate \( \pi \) as \( 3.14 \): \[ d^3 \approx \frac{240}{3.14} \approx 76.3945 \] Then take the cube root: \[ d \approx \sqrt[3]{76.3945} \approx 4.24 , \text{inches} \]

Step 6: Calculate \( r \) and \( h \)

Using \( d \): \[ r = \frac{d}{2} = \frac{4.24}{2} \approx 2.12 , \text{inches} \] And since \( h = d \): \[ h \approx 4.24 , \text{inches} \]

Step 7: Calculate the volume of the two hemispheres

The volume \( V_h \) of one hemisphere is given by: \[ V_h = \frac{2}{3} \pi r^3 \]

The volume of two hemispheres is: \[ V_{2h} = 2 \times \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3 \]

Calculating \( r^3 \): \[ r^3 \approx (2.12)^3 \approx 9.49 \]

Now substituting this value back for the volume of two hemispheres: \[ V_{2h} \approx \frac{4}{3} \pi (2.12^3) \approx \frac{4}{3} \cdot 3.14 \cdot 9.49 \] Calculating: \[ \approx \frac{4 \cdot 3.14 \cdot 9.49}{3} \approx \frac{118.750}{3} \approx 39.5833 , \text{in}^3 \]

Step 8: Calculate the total volume of the plastic container

The total volume \( V_t \) of the entire plastic container is: \[ V_t = V_c + V_{2h} \] Calculating this: \[ V_t \approx 60 + 39.5833 \approx 99.5833 , \text{in}^3 \]

Final Answer

Rounding to the nearest tenth: \[ V_t \approx 99.6 , \text{in}^3 \]

Thus, the volume of the entire plastic container is approximately 99.6 in³.

Answer 2

The National Oceanic and Atmospheric Administration is creating a buoy. They use a hemispherical piece for the bottom of the buoy and a cone piece for the top. The cone fits perfectly on the hemisphere bottom, with no overhang. The volume of the cone is approximately 183 cm 3
, and the height of the cone is equal to the diameter of the hemisphere. What is the volume of the entire plastic buoy? Round to the nearest tenths place if needed.

Answer 3

To find the volume of the entire buoy, we need to calculate the volume of the hemispherical part and the volume of the conical part, and then add the two volumes together.

Step 1: Understand the relationship between the cone and the hemisphere

Let:

• \( r \) be the radius of the hemisphere.
• The diameter of the hemisphere is \( 2r \).
• The height \( h \) of the cone is equal to the diameter of the hemisphere, which means \( h = 2r \).

Step 2: Volume of the cone

The volume \( V_c \) of the cone is given by the formula: \[ V_c = \frac{1}{3} \pi r^2 h \]

Since \( h = 2r \), we can substitute: \[ V_c = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]

We're given that the volume of the cone \( V_c \) is approximately \( 183 , \text{cm}^3 \): \[ \frac{2}{3} \pi r^3 = 183 \]

Step 3: Solve for \( r^3 \)

To isolate \( r^3 \), first multiply both sides by \( \frac{3}{2} \): \[ \pi r^3 = 183 \cdot \frac{3}{2} = 274.5 \]

Now divide by \( \pi \): \[ r^3 = \frac{274.5}{\pi} \]

Using \( \pi \approx 3.14 \): \[ r^3 \approx \frac{274.5}{3.14} \approx 87.5 \]

Step 4: Calculate \( r \)

Taking the cube root: \[ r \approx \sqrt[3]{87.5} \approx 4.43 , \text{cm} \]

Step 5: Volume of the hemisphere

The volume \( V_h \) of a hemisphere is given by the formula: \[ V_h = \frac{2}{3} \pi r^3 \]

Now we can substitute \( r^3 \): \[ V_h = \frac{2}{3} \pi (87.5) \] Calculating: \[ V_h = \frac{2}{3} \cdot \pi \cdot 87.5 \approx \frac{2}{3} \cdot 3.14 \cdot 87.5 \]

Calculating the numbers: \[ \approx \frac{2 \cdot 3.14 \cdot 87.5}{3} \approx \frac{548.712}{3} \approx 182.904 \]

Step 6: Total volume of the buoy

Now, add the volumes of the cone and the hemisphere: \[ V_t = V_h + V_c \] Substituting the values: \[ V_t \approx 182.904 + 183 \] Calculating: \[ V_t \approx 365.904 \]

Final Answer

Rounding to the nearest tenths place gives us: \[ V_t \approx 365.9 , \text{cm}^3 \]

Thus, the volume of the entire plastic buoy is approximately 365.9 cm³.

Answer 4

Using a cylinder and a sphere, Hagan is building a model of a tree. The volume of the sphere is 400 cm 3
. What is the volume of the cylinder if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter? Round to the nearest tenths place if needed.

Answer 5

To find the volume of the cylinder, we first need to determine the radius of the sphere, which will help us calculate the volume of the cylinder.

Step 1: Find the radius of the sphere

The volume \( V \) of a sphere is given by the formula: \[ V = \frac{4}{3} \pi r^3 \] where \( r \) is the radius of the sphere.

We know the volume of the sphere is \( 400 , \text{cm}^3 \): \[ \frac{4}{3} \pi r^3 = 400 \]

Step 2: Solve for \( r^3 \)

To isolate \( r^3 \), we first multiply both sides by \( \frac{3}{4} \): \[ \pi r^3 = 400 \cdot \frac{3}{4} = 300 \]

Now, divide both sides by \( \pi \): \[ r^3 = \frac{300}{\pi} \]

Approximating \( \pi \) as \( 3.14 \): \[ r^3 \approx \frac{300}{3.14} \approx 95.5 \]

Step 3: Calculate \( r \)

Taking the cube root to find \( r \): \[ r \approx \sqrt[3]{95.5} \approx 4.57 , \text{cm} \]

Step 4: Find the height of the cylinder

The height \( h \) of the cylinder is equal to the diameter of the sphere. The diameter \( d \) of the sphere is: \[ d = 2r \approx 2 \times 4.57 \approx 9.14 , \text{cm} \]

Step 5: Volume of the cylinder

The volume \( V_c \) of a cylinder is given by the formula: \[ V_c = \pi r^2 h \]

Substituting \( h = 2r \): \[ V_c = \pi r^2 (2r) = 2\pi r^3 \]

Now substituting \( r^3 \): \[ V_c = 2\pi \left(\frac{300}{\pi}\right) = 600 , \text{cm}^3 \]

Final Answer

The volume of the cylinder is: \[ \boxed{600 , \text{cm}^3} \]