Latest solutions
u84236168 ✎ Biotic factor - the impact of living organisms on each other. Abiotic factor is the effect of the inorganic environment on living organisms (chemical and physical). A) An increase in pressure is a physical factor, therefore, we classify it as abiotic. B) Earthquake is a physical abiotic factor. C) The epidemic is caused by microorganisms, therefore there is a biotic factor here. D) The interaction of wolves in a pack is a biotic factor. D) Competition between pines is a biotic factor, because Pines are living organisms. Answer: 11222 to the problem
u84236168 ✎ 1) The table shows that if there are more than 5 chicks in the nest, then the proportion of surviving chicks decreases sharply, therefore, we agree with this statement. 2) The death of the chicks is not explained in any way in the table, therefore, we cannot say anything about this statement. 3) Yes, the table shows that the fewer eggs in the clutch, the higher the care for the offspring, so the highest percentage of surviving chicks (100%) correlates with their smallest number (1), so we agree with this statement. 4) Regarding the fourth statement, we do not have any accurate information + the proportion of surviving chicks is decreasing, which means we do not agree with this statement. 5) The table does not contain information on what the number of eggs in a clutch is related to, therefore, we ignore this statement. Answer: 1, 3. to the problem
u84236168 ✎ A) Cactus spines and barberry spines are plant organs, an example is used in the comparative anatomical method of studying evolution. B) Remains are fossilized parts of ancient living beings, whose study is the science of paleontology, therefore, this is a paleontological method. C) Phylogenesis is the process of historical development of nature and individual organisms. In the phylogenetic series of a horse there may be its ancient ancestors, therefore, this is a paleontological method. D) Human multi-nipple refers to the comparative anatomical method, because the norm (two nipples) and atavism are compared. D) The appendix in humans is a rudiment, therefore, the norm and the rudiment are also compared here. Answer: 21122 to the problem
u84236168 ✎ 1) The speed cannot be directly proportional, otherwise, as the temperature decreases, the speed would strictly increase, which we do not observe on the graph. 2) The graph does not say anything about environmental resources, so we cannot say anything about this statement. 3) There is also no information about the genetic program on the graph, therefore, we cannot say anything. 4) The graph shows that the reproduction rate increases in the interval from 20 to 36 degrees, then we agree with this statement. 5) The graph shows that after 36 degrees the speed drops, which means we agree with this statement. Answer: 4, 5. to the problem
u84236168 ✎ In this picture, the external auditory canal, eardrum and cochlea (as can be seen from the shape) are correctly labeled. The remaining elements: 3 - chamber of the inner ear, 4 - hammer, 5 - incus. Answer: 1, 2, 6. to the problem
Hello! Another portion of problems with prisms, triangular prisms are considered. I combined several tasks that are similar in one “feature” - they have a section running through the middle line of the base. The questions are about calculating the surface area or volume of either the original prism or the cut one. What is important to remember here?
This property of similarity of figures relating to area, in particular about the triangle, was already discussed in one of the articles. But even if you suddenly forget this, the presented tasks are still intuitive and can be solved in one action.
77111. Through the midline of the base of a triangular prism, the lateral surface area of which is 6, a plane is drawn parallel to the lateral edge. Find the lateral surface area of the trimmed triangular prism.
It is said that the plane passes through the midline of the base, that is, through the points that are the midpoints of the adjacent sides of the triangle. Moreover, it runs parallel to the side edge - this means that the specified plane also passes through the midpoints of the corresponding adjacent sides of the other base.
Without any calculations, it is clear that the lateral surface area of the cut-off prism will be half that of the original one.
Look!
The prisms have a common height. The specified plane cuts two adjacent side faces in half.
Let's consider the third face (parallel to the section plane) - its surface area is also half as large, since the middle line of the triangle is half the size of the side of the triangle parallel to it.
Considering that the height remains unchanged (common for both prisms), we can conclude that the lateral surface area (the sum of the areas of all three faces) of the cut-off prism will be half as large.
Answer: 3
76147. A plane parallel to the side edge is drawn through the center line of the base of a triangular prism. The lateral surface area of the trimmed triangular prism is 20. Find the lateral surface area of the original prism.
The task is the opposite of the previous one. FFormula for the lateral surface area of a prism:
This means for a cut-off prism:
The height of the pyramids is common, so the area of the lateral surface of the original prism depends on the perimeter. Since the resulting triangles at the base of the prism are similar, and their corresponding sides are in the ratio 1:2, this means that the perimeter of the base of the original prism is twice the perimeter of the base of the cut-off one.
This means that the lateral surface area is also 2 times larger and equal to 40.
Answer: 40
27106. Through the middle line of the base of a triangular prism, the volume of which is 32, a plane is drawn parallel to the side edge. Find the volume of the cut-off triangular prism.
It is known that the volume of a prism is equal to the product of the area of the base and the height. The height for these prisms is common, which means the change in volume depends only on the change in surface area.
Let's consider the triangles lying at the bases of the prisms - they are similar. If we consider the base of the original prism relative to the base of the cut-off one, then the similarity coefficient will be equal to 2. What does this give us?
We know that the areas of similar figures are correlated as the square of the similarity coefficient, which means:
The base of the cut-off prism is 4 times smaller.
Thus, its volume will be 4 times smaller, that is, 8.
Formally, it can be written like this:
Answer: 8
74745. A plane parallel to the side edge is drawn through the center line of the base of a triangular prism. The volume of the cut-off triangular prism is 7. Find the volume of the original prism.
The task is the opposite of the previous one. The volume of the prism is equal to the product of the area of the base and the height:
The height is total, which means the volume changes depending on the change in the area of the base.
The triangle lying at the base of the original prism, as already said, is similar to the triangle lying at the base of the cut-off prism. The similarity coefficient is 2, since the section is drawn through the center line.
The areas of similar figures are related as the square of the similarity coefficient, that is:
Thus, the base area of the original prism is 4 times greater than the base area of the cut-off prism.
Therefore, the volume of the original prism will be 4 times greater than the volume of the cut-off prism. Thus, the required volume is 28.
Answer: 28
Three more problems about the surface area of a prism
245356. The surface area of a regular triangular prism is 6. What will be the surface area of the prism if all its edges are tripled?
Let's enlarge all the edges of the prism three times. What happens?
It turns out that each face of the resulting prism and the corresponding face of the original prism are similar figures. Moreover, the similarity coefficient is equal to 3. We know that the areas of similar figures are proportional to the square of the similarity coefficient, that is:
This means that the area of each individual face of our prism will increase by 9 times. Since the surface area of the entire prism is the sum of the areas of all the faces, it goes without saying that the entire surface area of the prism will also increase by 9 times.
Answer: 54
*In fact, it doesn’t matter what kind of body we are talking about (a prism, a pyramid, a cube, a parallelepiped), the essence is the same.
In a triangular prism, the two side faces are perpendicular. Their common edge is 30 and is separated from the other lateral edges by 3 and 4. Find the lateral surface area of this prism.
Latest solutions
u84236168 ✎ Biotic factor - the impact of living organisms on each other. Abiotic factor is the effect of the inorganic environment on living organisms (chemical and physical). A) An increase in pressure is a physical factor, therefore, we classify it as abiotic. B) Earthquake is a physical abiotic factor. C) The epidemic is caused by microorganisms, therefore there is a biotic factor here. D) The interaction of wolves in a pack is a biotic factor. D) Competition between pines is a biotic factor, because Pines are living organisms. Answer: 11222 to the problem
u84236168 ✎ 1) The table shows that if there are more than 5 chicks in the nest, then the proportion of surviving chicks decreases sharply, therefore, we agree with this statement. 2) The death of the chicks is not explained in any way in the table, therefore, we cannot say anything about this statement. 3) Yes, the table shows that the fewer eggs in the clutch, the higher the care for the offspring, so the highest percentage of surviving chicks (100%) correlates with their smallest number (1), so we agree with this statement. 4) Regarding the fourth statement, we do not have any accurate information + the proportion of surviving chicks is decreasing, which means we do not agree with this statement. 5) The table does not contain information on what the number of eggs in a clutch is related to, therefore, we ignore this statement. Answer: 1, 3. to the problem
u84236168 ✎ A) Cactus spines and barberry spines are plant organs, an example is used in the comparative anatomical method of studying evolution. B) Remains are fossilized parts of ancient living beings, whose study is the science of paleontology, therefore, this is a paleontological method. C) Phylogenesis is the process of historical development of nature and individual organisms. In the phylogenetic series of a horse there may be its ancient ancestors, therefore, this is a paleontological method. D) Human multi-nipple refers to the comparative anatomical method, because the norm (two nipples) and atavism are compared. D) The appendix in humans is a rudiment, therefore, the norm and the rudiment are also compared here. Answer: 21122 to the problem
u84236168 ✎ 1) The speed cannot be directly proportional, otherwise, as the temperature decreases, the speed would strictly increase, which we do not observe on the graph. 2) The graph does not say anything about environmental resources, so we cannot say anything about this statement. 3) There is also no information about the genetic program on the graph, therefore, we cannot say anything. 4) The graph shows that the reproduction rate increases in the interval from 20 to 36 degrees, then we agree with this statement. 5) The graph shows that after 36 degrees the speed drops, which means we agree with this statement. Answer: 4, 5. to the problem
u84236168 ✎ In this picture, the external auditory canal, eardrum and cochlea (as can be seen from the shape) are correctly labeled. The remaining elements: 3 - chamber of the inner ear, 4 - hammer, 5 - incus. Answer: 1, 2, 6. to the problem
Task 9 (Probe-2015, profile level)
The volume of a regular icosahedron is \(\frac(144(3+\sqrt5))(25).\) Find the length of its edge.
Solution
An icosahedron is a regular convex polyhedron. Each of the 20 faces is an equilateral triangle. The volume of the icosahedron can be calculated using the formula:
$$V = \frac5(12)(3+\sqrt5)a^3.$$
Then, $$\frac5(12)(3+\sqrt5)a^3 = \frac(144(3+\sqrt5))(25),$$
$$a^3 = \frac(12^3)(5^3),$$
$$a = 12/5 = 2.4.$$
Answer: 2.4.
Task 12 (Early Unified State Exam - 2015, profile level)
A plane parallel to the side edge is drawn through the middle line of the base of the triangular prism. The lateral surface area of the cut-off triangular prism is 43. Find the lateral surface area of the original prism.
Solution
Let the side of the base of the original prism be equal to a, and the height equal to h. Then the area of its lateral surface is:
For a cut off triangular prism, the side of the base will be equal to a/2, and the height, like that of a large prism, will be equal to h. Then the area of its lateral surface will be equal to:
$$S_2=3ah/2 = 1.5ah.$$
Then \(1.5ah = 43\), hence \(3ah = 86\).
This means that the lateral surface area of the original prism is 86.
Task 13 (Early Unified State Examination - 2015, basic level)
Two cylindrical circles are given. The first mug is one and a half times lower than the second, and the second is twice as wide as the first. How many times is the volume of the second mug greater than the volume of the first?
A plane parallel to the side edge is drawn through the middle line of the base of the triangular prism. The volume of the cut-off triangular prism is 5. Find the volume of the original prism.
A plane parallel to the side edge is drawn through the middle line of the base of the triangular prism. The lateral surface area of the trimmed triangular prism is 8. Find the lateral surface area of the original prism.
Find the lateral surface area of a regular hexagonal prism whose base side is 5 and height is 10.
69) No. 27084_ Find the volume of a regular hexagonal prism, the sides of the base are equal to 1, and the side edges are equal to .
70) No. 245357_ Find the volume of a regular hexagonal prism, all edges of which are equal.
71) No. 27108_ Find the volume of a prism, the bases of which contain regular hexagons with sides of 2, and the side edges are equal and inclined to the plane of the base at an angle of 30.
The sides of the base of a regular quadrangular pyramid are equal to 10, the side edges are equal to 13. Find the surface area of this pyramid.
In a regular quadrangular pyramid, the height is 6 and the side edge is 10. Find its volume.
Find the surface area of a regular quadrangular pyramid whose base sides are 6 and whose height is 4.
Find the lateral surface area of a regular quadrangular pyramid whose base side is 6 and height is 4.
In a regular quadrangular pyramid, the height is 12 and the volume is 200. Find the side edge of this pyramid.
The base of the pyramid is a rectangle with sides 3 and 4. Its volume is 16. Find the height of this pyramid.
78) No. 27110_ The base of the pyramid is a rectangle, one side face is perpendicular to the base plane, and the other three side faces are inclined to the base plane at an angle of 60. The height of the pyramid is 6. Find the volume of the pyramid.
Find the volume of a pyramid whose height is 6 and whose base is a rectangle with sides 3 and 4.
80) No. 27087_ Find the volume of a regular triangular pyramid whose sides of the base are equal to 1 and whose height is equal to .
81) No. 27088_ Find the height of a regular triangular pyramid whose base sides are equal to 2 and whose volume is equal to .
The lateral edges of a triangular pyramid are mutually perpendicular, each of them is equal to 3. Find the volume of the pyramid.
The edges of a tetrahedron are equal to 1. Find the cross-sectional area passing through the midpoints of its four edges.