Mathematics Answer Note #48

Question 1 From the given cylinder, determine its height. Its volume is 1000 cm cubed and use = 3.142. Answer To determine the height of the given cylinder, we will first need to utilize the formula for the volume of a cylinder, which is V = πr^2h, where V is the volume, π is the mathematical constant approximately equal to 3.142, r is the radius of the circular base, and h is the height. We also know that the volume of the given cylinder is 1000 cm cubed, so we can plug in the values in the formula and solve for h. V = πr^2h 1000 = 3.142(r^2)h Next, we need to take into consideration the fact that we are given the volume in cubic centimeters (cm cubed), but the formula requires the radius to be in centimeters (cm) as well. This means we need to convert the given volume from cm cubed to cm by taking the cube root of the volume value. ∛ 1000 = 10 cm Now, we can substitute the radius value of 10 cm into the formula: 1000 = 3.142(10^2)h 1000 = 3.142(100)h 1000 = 314.2h To isolate h, we divide both sides by 314.2: 1000/314.2 = h h ≈ 3.183 cm Therefore, the height of the given cylinder is approximately 3.183 cm. Question 2 The volume of the given cylinder is 2 400cm3. Its height is 30cm. What is the radius of this cylinder? Use = 3.142. Answer According to the formula for the volume of a cylinder, V = πr^2h, where V is the volume, r is the radius, and h is the height, we can rearrange this formula to solve for the radius, which would be r = √(V/πh). Given that the volume (V) is 2,400cm^3, and the height (h) is 30cm, we can plug these values into the formula to solve for the radius: r = √(2,400cm^3/3.142*30cm) = √25.45cm = 5.05cm Therefore, the radius of the given cylinder is 5.05cm. Question 3 Is the following statement True or False? I have a sphere whose volume is 300 cm2. I also have a cube that has the same volume. The surface areas of the two objects are the same. Answer The statement "I have a sphere whose volume is 300 cm2. I also have a cube that has the same volume. The surface areas of the two objects are the same" is false. According to the formula for calculating the volume of a sphere and a cube, the volume of a sphere is (4/3)*π*r^3 and the volume of a cube is s^3, where r is the radius of the sphere and s is the length of the side of the cube. Given the information that the volume of both objects is 300 cm2, we can set up the following equations: (4/3)*π*r^3 = 300 cm2 s^3 = 300 cm2 To find the surface area of a sphere and a cube, we use the formulas 4*π*r^2 and 6*s^2, respectively. Using the volume equations, we can solve for the radius of the sphere and the side length of the cube: r = (3/4)*300/(π)^(1/3) ≈ 5.82 cm s = 300^(1/3) ≈ 6.16 cm Plugging these values into the surface area formulas, we get: Surface area of sphere = 4*π*(5.82)^2 ≈ 425.93 cm2 Surface area of cube = 6*(6.16)^2 ≈ 226.26 cm2 As we can see, the surface areas of the two objects are not the same. The surface area of the sphere is approximately double the surface area of the cube. Therefore, the statement is false. Question 4 A room is 8.7m long, 8.3m wide and 7.6m high, what is its volume? Answer The volume of the room can be calculated by multiplying its length, width, and height. Therefore, the volume of the room would be 8.7m x 8.3m x 7.6m = 672.54 cubic meters. To further break down the calculation, we can use the formula for finding the volume of a rectangular prism, which is length x width x height. In this case, the length is 8.7m, the width is 8.3m, and the height is 7.6m. Substituting these values into the formula, we get 8.7m x 8.3m x 7.6m = 672.54 cubic meters. In order for this calculation to be accurate, it is important to note that all measurements must be in the same unit, which in this case is meters. This ensures that the calculation is done correctly and the resulting volume is in the appropriate unit. Question 5 wHAT IS THE AREA OF A CIRCLE WITH A RADIUS OF 10CM? Answer The area of a circle is a common calculation that is used in many subjects, including mathematics, physics, and even engineering. It is important to understand this concept as it can be applied to various real-life situations, such as calculating the area of a circular pool or the surface area of a spherical object. According to the formula, the area of a circle is equal to π multiplied by the square of the radius. Therefore, the area of a circle with a radius of 10cm can be calculated as follows: A = πr² A = π(10cm)² A = 100π cm² Thus, the area of a circle with a radius of 10cm is approximately 314.16 cm². It is worth noting that the number π represents the ratio of the circumference of any circle to its diameter. Its decimal approximation is 3.14159. For more accurate calculations, the value of π can be taken as 3.14 or used in its exact form for more precise results. To further understand the concept of the area of a circle, it is useful to refer to academic sources. In mathematics, the area of a circle is classified as a two-dimensional measurement known as the circular region. This concept is extensively discussed in various algebra, calculus, and geometry textbooks. In the field of physics, the area of a circle is often used in the calculation of the circumference of a circle, which can ultimately lead to the determination of the speed, velocity, and acceleration of physical objects in circular motion. This topic is tackled indepth in the book "Essential Physics for Scientists and Engineers" by Douglas Giancoli. Furthermore, the area of a circle has practical applications in engineering, especially in the design and measurement of circular structures. The book "Engineering Mechanics: Dynamics" by Russell Hibbeler explains the importance of this calculation in detail. In conclusion, the area of a circle with a radius of 10cm is 314.16 cm². This calculation is derived from the fundamental formula of A = πr² and is widely used in mathematics, physics, and engineering. By referring to various academic sources, we can further understand the significance of this concept and how it is applied in different fields of study.