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Magazine Chapter 0: Problem 31
(a) Compute the circumference and area of a circle of radius \(0.12 \mathrm{~m}\). (b) Compute the surface area and volume of a sphere of radius \(0.21 \mathrm{~m}\). (c) Compute the total surface area and volume of a rectangular solid of length \(0.18 \mathrm{~m},\) width \(0.15 \mathrm{~m},\) and height \(0.8 \mathrm{~m} .\) (d) Compute the total surface area and volume of a cylinder of radius \(0.18 \mathrm{~m}\) and height \(0.33 \mathrm{~m}\).
Short Answer
Expert verified
Circle: C = 0.754 m, A = 0.01445 m虏; Sphere: A = 0.5541 m虏, V = 0.0388 m鲁; Rectangular Solid: A = 0.522 m虏, V = 0.0216 m鲁; Cylinder: A = 0.5788 m虏, V = 0.0336 m鲁.
Step by step solution
01
Compute the Circumference of the Circle
The circumference of a circle is given by the formula \( C = 2\pi r \), where \( r \) is the radius of the circle. For a radius of \( 0.12 \text{ m} \), the circumference is \( C = 2 \pi \times 0.12 \text{ m} = 0.24\pi \text{ m} \approx 0.754 \text{ m} \).
02
Compute the Area of the Circle
The area of a circle is given by the formula \( A = \pi r^2 \). Substituting \( r = 0.12 \text{ m} \) into the formula gives \( A = \pi \times (0.12)^2 \approx 0.01445 \text{ m}^2 \).
03
Compute the Surface Area of the Sphere
The surface area of a sphere is given by \( A = 4\pi r^2 \), where \( r \) is the radius. With a radius of \( 0.21 \text{ m} \), the surface area is \( A = 4\pi \times (0.21)^2 \approx 0.5541 \text{ m}^2 \).
04
Compute the Volume of the Sphere
The volume of a sphere is calculated using the formula \( V = \frac{4}{3}\pi r^3 \). Substituting \( r = 0.21 \text{ m} \), we find \( V = \frac{4}{3}\pi \times (0.21)^3 \approx 0.0388 \text{ m}^3 \).
05
Compute the Surface Area of the Rectangular Solid
The surface area of a rectangular solid is calculated by \( A = 2(lw + lh + wh) \). With \( l = 0.18 \text{ m}, w = 0.15 \text{ m}, h = 0.8 \text{ m} \), we have \( A = 2(0.18 \times 0.15 + 0.18 \times 0.8 + 0.15 \times 0.8) = 0.522 \text{ m}^2 \).
06
Compute the Volume of the Rectangular Solid
The volume of a rectangular solid is found using \( V = lwh \). Substitute the given dimensions \( l = 0.18 \text{ m}, w = 0.15 \text{ m}, h = 0.8 \text{ m} \) to get \( V = 0.18 \times 0.15 \times 0.8 \approx 0.0216 \text{ m}^3 \).
07
Compute the Surface Area of the Cylinder
The surface area of a cylinder is given by \( A = 2\pi r(h + r) \). For \( r = 0.18 \text{ m} \) and \( h = 0.33 \text{ m} \), the surface area is \( A = 2\pi \times 0.18(0.33 + 0.18) \approx 0.5788 \text{ m}^2 \).
08
Compute the Volume of the Cylinder
The volume of a cylinder is calculated with \( V = \pi r^2 h \). Substituting \( r = 0.18 \text{ m} \) and \( h = 0.33 \text{ m} \), we find \( V = \pi \times (0.18)^2 \times 0.33 \approx 0.0336 \text{ m}^3 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Circumference of a Circle
One of the most fundamental concepts in geometry is the circumference of a circle. It's essentially the total length around the circle, similar to the perimeter of a polygon. The formula used is \( C = 2\pi r \), where \( \pi \approx 3.14159 \) and \( r \) is the radius of the circle. Understanding this formula is quite simple.
- The number \( 2\pi \) represents the ratio of the circumference to the diameter of any circle.
- Multiplying \( 2\pi \) by the radius gives you the total distance around the circle.
Surface Area of a Sphere
The surface area of a sphere is another crucial geometrical measurement. It determines the total area covering the exterior of a sphere. The formula is \( A = 4\pi r^2 \), where \( r \) is the sphere's radius. This formula is derived because a sphere has no edges, just like a circle has no sides.
- \( 4\pi \) can be seen as the circle's area \( \pi r^2 \) times four, enveloping the sphere completely.
- This makes it handy for finding out how much material you would need to cover a sphere.
Volume of a Cylinder
The volume of a cylinder is key in understanding how much space it occupies. This is particularly useful in contexts such as filling tanks or pipes. The formula for volume of a cylinder is \( V = \pi r^2 h \), where \( r \) is the radius of the circular base, and \( h \) is the height of the cylinder.
- The base area, \( \pi r^2 \), is multiplied by the height to account for the cylinder's vertical dimension.
- It's like stacking circles up to a certain height to fill the space.
Rectangular Solid Volume
The volume of a rectangular solid is a vital aspect in geometry, especially while dealing with boxes, houses, or any cuboid-shaped object. The volume determines how much space is inside the solid and is calculated using \( V = lwh \). This formula stands for multiplying the
- length \( l \)
- width \( w \)
- height \( h \)
Surface Area of a Cylinder
The surface area of a cylinder is an interesting geometrical concept that calculates how much area covers the outside of a cylinder. This is useful for knowing how much material is needed to wrap or coat a cylinder. The formula for this is \( A = 2\pi r(h + r) \), where \( r \) is the radius and \( h \) is the height.
- The component \( 2\pi rh \) calculates the side area (the rectangle if it were unrolled).
- The part \( 2\pi r^2 \) represents the areas of the two bases.
