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A water bottle has a height of 16 inches and a radius of 4 inches. What is the volume, in cubic inches, of the water bottle? Use 3.14 for pi.
To find the volume of a cylindrical water bottle, you can use the formula for the volume of a cylinder:
\[ V = \pi r^2 h \]
where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.
Given:
- Height (\( h \)) = 16 inches
- Radius (\( r \)) = 4 inches
- \(\pi \approx 3.14\)
Now, plug the values into the formula:
\[ V = 3.14 \times (4)^2 \times 16 \]
First, calculate \( (4)^2 \):
\[ (4)^2 = 16 \]
Now substitute this back into the volume formula:
\[ V = 3.14 \times 16 \times 16 \]
Now calculate \( 16 \times 16 \):
\[ 16 \times 16 = 256 \]
Now substitute this back into the volume formula:
\[ V = 3.14 \times 256 \]
Finally, calculate that:
\[ V \approx 3.14 \times 256 = 804.44 \]
Therefore, the volume of the water bottle is approximately 804.44 cubic inches.
Find the volume, in cubic inches, of a cone with a radius of 13 inches and a height of 27 inches. Round your answer to the nearest hundredth. Use 3.14 for pi.
To find the volume of a cone, you can use the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where:
- \( V \) is the volume,
- \( r \) is the radius,
- \( h \) is the height.
Given:
- Radius (\( r \)) = 13 inches
- Height (\( h \)) = 27 inches
- \(\pi \approx 3.14\)
Now, plug the values into the formula:
\[ V = \frac{1}{3} \times 3.14 \times (13)^2 \times 27 \]
First, calculate \( (13)^2 \):
\[ (13)^2 = 169 \]
Now substitute this back into the volume formula:
\[ V = \frac{1}{3} \times 3.14 \times 169 \times 27 \]
Now calculate \( 169 \times 27 \):
\[ 169 \times 27 = 4563 \]
Now substitute this back into the volume formula:
\[ V = \frac{1}{3} \times 3.14 \times 4563 \]
Next, multiply \( 3.14 \times 4563 \):
\[ 3.14 \times 4563 \approx 14358.42 \]
Now, divide by 3:
\[ V \approx \frac{14358.42}{3} \approx 4786.14 \]
Therefore, the volume of the cone is approximately 4786.14 cubic inches when rounded to the nearest hundredth.
A cone-shaped container on a farm is used for storing animal feed. The container has a radius of 4 feet and a height of 12 feet. What volume of feed, in cubic feet, can the container store? Round your answer to the nearest hundredth
To find the volume of a cone-shaped container, you can use the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where:
- \( V \) is the volume,
- \( r \) is the radius,
- \( h \) is the height.
Given:
- Radius (\( r \)) = 4 feet
- Height (\( h \)) = 12 feet
- Using \( \pi \approx 3.14\)
Now, plug in the values into the formula:
\[ V = \frac{1}{3} \times 3.14 \times (4)^2 \times 12 \]
First, calculate \( (4)^2 \):
\[ (4)^2 = 16 \]
Now substitute this back into the volume formula:
\[ V = \frac{1}{3} \times 3.14 \times 16 \times 12 \]
Now calculate \( 16 \times 12 \):
\[ 16 \times 12 = 192 \]
Substituting this back into the volume formula gives us:
\[ V = \frac{1}{3} \times 3.14 \times 192 \]
Now multiply \( 3.14 \times 192 \):
\[ 3.14 \times 192 \approx 602.88 \]
Now, divide by 3:
\[ V \approx \frac{602.88}{3} \approx 200.96 \]
Thus, the volume of feed that the cone-shaped container can store is approximately 200.96 cubic feet, rounded to the nearest hundredth.
Explain the diameter of a sphere
The diameter of a sphere is defined as the straight line segment that passes through the center of the sphere and connects two points on its surface. In other words, it is the longest distance across the sphere, effectively cutting it into two equal halves.
Key Points about the Diameter of a Sphere:
-
Relation to Radius: The diameter (\(d\)) is always twice the radius (\(r\)) of the sphere. The radius is the distance from the center of the sphere to any point on its surface, so: \[ d = 2r \]
-
Units: The diameter is measured in linear units, such as centimeters, inches, meters, etc., and it is the same unit of measurement as the radius.
-
Geometric Significance: The diameter is a crucial measurement in geometry, as it helps to define other properties of the sphere:
- Circumference: For a sphere, the circumference of a great circle (the largest circle that can be drawn on the surface) can be calculated using the diameter: \[ C = \pi d \]
- Surface Area and Volume: The diameter is also used in the formulas for calculating the surface area and volume of a sphere:
- Surface Area: \[ A = 4\pi r^2 = \pi d^2 \]
- Volume: \[ V = \frac{4}{3} \pi r^3 = \frac{\pi d^3}{6} \]
-
Position: The diameter is a specific type of chord (a line segment connecting two points on the sphere) that passes through the center. Therefore, it is also the longest possible chord in the sphere.
Understanding the diameter of a sphere is fundamental in various fields, including geometry, physics, engineering, and architecture, as it relates to the size and capacity of spherical objects.
The diameter of a sphere
* is the longest distance on a sphere
*is half the length of a radius.
*does not pass through the center of the sphere
*is the distance from the center of the sphere to any point on the surface of the sphere.
* is the longest distance on a sphere
*is half the length of a radius.
*does not pass through the center of the sphere
*is the distance from the center of the sphere to any point on the surface of the sphere.
Let's analyze each of the statements you've provided about the diameter of a sphere to determine which ones are correct:
-
Is the longest distance on a sphere:
- True. The diameter is indeed the longest straight line that can be drawn across a sphere, connecting two points on its surface while passing through the center.
-
Is half the length of a radius:
- False. This statement is incorrect. The diameter is actually twice the length of the radius, not half. The relationship is:
\[ d = 2r \]
- False. This statement is incorrect. The diameter is actually twice the length of the radius, not half. The relationship is:
-
Does not pass through the center of the sphere:
- False. The diameter must pass through the center of the sphere. By definition, it is the line segment that connects two points on the sphere's surface while going through its center.
-
Is the distance from the center of the sphere to any point on the surface of the sphere:
- False. This description actually defines the radius of the sphere, not the diameter. The radius is the distance from the center to any point on the surface, while the diameter is the distance across the sphere, passing through the center.
Summary:
- The only correct statement is that the diameter of a sphere is the longest distance on a sphere. The other statements contain inaccuracies regarding the definitions and relationships involving the radius and diameter.