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Answer 1

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Answer 2

Find the surface area of a hemisphere with a radius 10m. Leave your answer in terms of pi.

Answer 3

The surface area \( A \) of a hemisphere can be calculated using the formula:

\[ A = 2\pi r^2 + \pi r^2 \]

Here, \( 2\pi r^2 \) represents the curved surface area and \( \pi r^2 \) is the area of the circular base.

Given that the radius \( r \) is 10 m, we substitute this value into the formula:

\[ A = 2\pi (10)^2 + \pi (10)^2 \]

Calculating the areas:

\[ A = 2\pi (100) + \pi (100) \] \[ A = 200\pi + 100\pi \] \[ A = 300\pi \]

Thus, the surface area of the hemisphere is \( 300\pi , \text{m}^2 \).

Answer 4

The surface area of a sphere is 7.84pi ft^2. What is the radius of thissphere?

Answer 5

The surface area \( A \) of a sphere can be calculated using the formula:

\[ A = 4\pi r^2 \]

Given that the surface area is \( 7.84\pi , \text{ft}^2 \), we can set up the equation:

\[ 4\pi r^2 = 7.84\pi \]

To simplify, we can divide both sides by \( \pi \):

\[ 4r^2 = 7.84 \]

Now, divide both sides by 4:

\[ r^2 = \frac{7.84}{4} = 1.96 \]

Now, take the square root of both sides to find \( r \):

\[ r = \sqrt{1.96} \]

Calculating the square root:

\[ r = 1.4 , \text{ft} \]

Thus, the radius of the sphere is \( 1.4 , \text{ft} \).

Answer 6

How much ice cream is in a spherical scoop of ice cream with a radius of 3 cm? Use 3.14 as an approximation for pi. Round your answer to the nearest whole number.

Answer 7

The volume \( V \) of a sphere can be calculated using the formula:

\[ V = \frac{4}{3} \pi r^3 \]

Given that the radius \( r \) is 3 cm, we can substitute this value into the formula. We'll use \( \pi \approx 3.14 \).

\[ V = \frac{4}{3} \times 3.14 \times (3)^3 \]

Calculating \( (3)^3 \):

\[ (3)^3 = 27 \]

Now, substitute this back into the formula:

\[ V = \frac{4}{3} \times 3.14 \times 27 \]

Calculating \( \frac{4}{3} \times 27 \):

\[ \frac{4 \times 27}{3} = \frac{108}{3} = 36 \]

Now substituting this back:

\[ V = 36 \times 3.14 \]

Calculating this:

\[ V = 113.04 \]

Rounding to the nearest whole number gives:

\[ V \approx 113 , \text{cm}^3 \]

So, the volume of the spherical scoop of ice cream is approximately 113 cm³.

Answer 8

Suppose a sphere, cylinder, and cone all share the same radius and the cylinder and cone also share the same height, which is twice the radius. Which of the following is true about the relationship between the volume formulas for the sphere, cylinder, and cone

Answer 9

Let's derive the volume formulas for the sphere, cylinder, and cone based on the given conditions and compare them.

1. Volume of the Sphere: The formula for the volume \( V \) of a sphere with radius \( r \) is:

   \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]

2. Volume of the Cylinder: The formula for the volume \( V \) of a cylinder with radius \( r \) and height \( h \) is:

   \[ V_{\text{cylinder}} = \pi r^2 h \]

   Given that the height \( h \) is twice the radius, we have:

   \[ h = 2r \]

   Substituting this into the formula gives:

   \[ V_{\text{cylinder}} = \pi r^2 (2r) = 2\pi r^3 \]

3. Volume of the Cone: The formula for the volume \( V \) of a cone with radius \( r \) and height \( h \) is:

   \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]

   Again, substituting \( h = 2r \) gives:

   \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]

Summary of the Volumes

• Volume of the Sphere: \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]

• Volume of the Cylinder: \[ V_{\text{cylinder}} = 2\pi r^3 \]

• Volume of the Cone: \[ V_{\text{cone}} = \frac{2}{3} \pi r^3 \]

Relationship Between the Volumes

Now we can express the volume relationships:

• The volume of the cylinder is: \[ V_{\text{cylinder}} = 2\pi r^3 \]

• The volume of the sphere can be compared to the volume of the cylinder: \[ V_{\text{cylinder}} \text{ is larger than } V_{\text{sphere}} \text{ since } 2\pi r^3 > \frac{4}{3} \pi r^3. \]

• The volume of the cone can be compared as well: \[ V_{\text{cone}} = \frac{2}{3} \pi r^3, \text{ which is less than } V_{\text{sphere}} \text{ since } \frac{2}{3} \pi r^3 < \frac{4}{3} \pi r^3. \]

Conclusion

So the relationships can be summarized as follows:

1. \( V_{\text{sphere}} > V_{\text{cone}} \)
2. \( V_{\text{cylinder}} > V_{\text{sphere}} \)
3. \( V_{\text{cylinder}} > V_{\text{cone}} \)

In summary:

• The sphere has a volume greater than the cone.
• The cylinder has a volume greater than both the sphere and the cone.

Answer 10

The volume of a sphere is 48m^3. What is the volume of a cylinder its radius is the same as the sphere’s and its height is equal to the sphere’s diameter?

Answer 11

To find the volume of the cylinder, we first need to gather information about the sphere. The volume \( V \) of a sphere is given by the formula:

\[ V = \frac{4}{3} \pi r^3 \]

Step 1: Find the radius of the sphere

Given that the volume of the sphere is \( 48 , \text{m}^3 \), we can set up the equation:

\[ \frac{4}{3} \pi r^3 = 48 \]

To isolate \( r^3 \), we first multiply both sides by \( \frac{3}{4} \):

\[ \pi r^3 = 48 \times \frac{3}{4} = 36 \]

Next, divide by \( \pi \):

\[ r^3 = \frac{36}{\pi} \]

Taking the cube root gives us the radius of the sphere:

\[ r = \sqrt[3]{\frac{36}{\pi}} \]

Step 2: Find the diameter of the sphere

The diameter \( d \) of the sphere is:

\[ d = 2r \]

Step 3: Find the volume of the cylinder

The formula for the volume \( V \) of a cylinder is:

\[ V = \pi r^2 h \]

where \( h \) is the height of the cylinder. Since the height of the cylinder is equal to the diameter of the sphere:

\[ h = d = 2r \]

Now, substitute \( h \) into the volume formula for the cylinder:

\[ V = \pi r^2 (2r) = 2\pi r^3 \]

Step 4: Substitute \( r^3 \) to find the volume of the cylinder

We previously found that:

\[ r^3 = \frac{36}{\pi} \]

Now substitute this into the cylinder volume equation:

\[ V = 2\pi \left( \frac{36}{\pi} \right) = 2 \times 36 = 72 , \text{m}^3 \]

Conclusion

The volume of the cylinder is \( 72 , \text{m}^3 \).