The three sides of a triangle are 4,5 and 6.show that cosine of the largest angle is 1/8.hence find the area of the triangle.
To find the cosine of the largest angle, let's use the Law of Cosines, which states that for a triangle with sides a, b, and c and opposite angles A, B, and C respectively, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, let's label the sides as follows:
a = 4
b = 5
c = 6
Let's solve for the cosine of the largest angle, which is opposite side c:
36 = 16 + 25 - 40 * cos(C)
Rearrange the equation:
40 * cos(C) = 41 - 36
40 * cos(C) = 5
Divide both sides by 40:
cos(C) = 1/8
Now, let's find the area of the triangle using the formula:
Area = (1/2) * base * height
We can use Heron's formula to find the area of the triangle, which states that the area can be calculated using the semi-perimeter (s) and the lengths of the sides (a, b, and c) as follows:
Area = sqrt(s * (s - a) * (s - b) * (s - c))
Let's calculate the semi-perimeter:
s = (a + b + c)/2
s = (4 + 5 + 6)/2
s = 15/2 = 7.5
Now, plug in the values into the formula:
Area = sqrt(7.5 * (7.5 - 4) * (7.5 - 5) * (7.5 - 6))
= sqrt(7.5 * 3.5 * 2.5 * 1.5)
= sqrt(82.125)
≈ 9.06
Therefore, the area of the triangle is approximately 9.06 square units.
what a lot of work!
Ok, we found that cosC = 1/8
That means that sinC = √63/8
Now, the area of the triangle is
A = 1/2 ab sinC = 1/2 * 4*5*√63/8 = 5/2 √63
Yes, you are correct. I apologize for the extensive explanation. The area of the triangle can indeed be found using the formula A = 1/2 * ab * sin(C).
By substituting the given values, we have:
A = 1/2 * 4 * 5 * √63/8
= 5/2 * √63/8
= 5/2 * (√63/√64)
= 5/2 * (√63/8)
= 5/2 * (√63/2√2)
= 5/4 * √63
Therefore, the area of the triangle is (5/4) * √63 square units.
Which behavior is both territorial and used for courtship?
sparring or wrestling using antlers or horns
dance rituals with potential partners
defending a nest from predators
baring teeth as a sign of aggression
The behavior that is both territorial and used for courtship is "dance rituals with potential partners."
Birds display a wide variety of courtship behaviors. Which behavior is also a territorial behavior?
singing a particular song to help potential mates hear them
flashing feathers to show their health and grooming abilities
Dancing and calling on a lek (communal ground) with other males, competing for female attention
creating a bower (arch) with interesting objects and defending it from rival males
The behavior that is both a courtship behavior and a territorial behavior is "creating a bower (arch) with interesting objects and defending it from rival males."
Why is climate change a concern for animal reproductive behavior?
Animals might need to spend more time hunting, foraging, and migrating.
Animals might have to change their courtship behavior due to climate change.
Climate change might genetically alter organisms and decrease their reproductive capabilities.
Climate change might reduce or destroy reproductive habitats, eliminating breeding grounds.
All of the listed options are valid reasons why climate change is a concern for animal reproductive behavior.
1. Animals might need to spend more time hunting, foraging, and migrating: Climate change can lead to shifts in temperature, rainfall patterns, and availability of resources, which can affect the availability and distribution of food sources for animals. This can disrupt their reproductive cycles and energy availability for reproduction.
2. Animals might have to change their courtship behavior due to climate change: Changes in climate, such as altered temperature or rainfall patterns, can affect the timing and availability of resources needed for courtship rituals. This can lead to changes in the timing or nature of courtship behavior, affecting successful breeding and reproductive success.
3. Climate change might genetically alter organisms and decrease their reproductive capabilities: Climate change can lead to genetic changes in populations due to natural selection and adaptation to new environmental conditions. These genetic changes can potentially affect reproductive capabilities, fertility, and overall fitness of the species.
4. Climate change might reduce or destroy reproductive habitats, eliminating breeding grounds: Climate change can result in the loss, degradation, or fragmentation of habitats necessary for breeding and reproduction. This can lead to the loss of suitable environments, nesting sites, mating areas, or migration routes, impacting the ability of animals to reproduce successfully.
How does hibernation help animals survive?
Hibernation makes access to food easier.
Hibernation increases the chances of meeting a mate.
Hibernation protects the animal young from predators.
Hibernation reduces the energy required for survival.
The correct answer is: Hibernation reduces the energy required for survival.
During hibernation, animals enter a state of dormancy where their metabolic rate decreases significantly. This results in a lowered body temperature, heart rate, and breathing rate. By reducing the energy required for basic bodily functions, hibernating animals can conserve energy and survive periods of food scarcity or harsh environmental conditions, such as winter.
While hibernating, animals heavily rely on the fat stores accumulated prior to their dormant state. They do not actively search for food, which is why hibernation does not make access to food easier.
Hibernation is not specifically related to meeting a mate, although some animals may emerge from hibernation during the mating season.
Additionally, protection of young from predators is not a direct benefit of hibernation. In some cases, animals may give birth while in hibernation, but this is not a universal attribute of all hibernating species.
Why do courtship behaviors usually happen in response to seasonal changes?
to ensure that offspring do not interfere with migration season
to ensure that offspring are born in optimal survival conditions
to ensure that mating does not interfere with migration season
to ensure that mating happens in optimal survival conditions
The correct answer is: to ensure that mating happens in optimal survival conditions.
Courtship behaviors usually occur in response to seasonal changes to ensure that mating happens at a time when environmental conditions are optimal for the survival and successful reproduction of offspring.
By timing their courtship behaviors with specific seasons, animals can take advantage of favorable conditions such as abundant food resources, suitable nesting or mating sites, or optimal weather conditions. This increases the chances of successful reproduction and the survival of their offspring.
Timing courtship behaviors with seasonal changes also helps to synchronize mating efforts among individuals of the same species, maximizing the chances of successful mate selection and reproductive success.
While migration season and offspring interference can be factors in certain cases, the primary reason for courtship behaviors aligning with seasonal changes is to ensure that mating occurs in optimal survival conditions.
Which factors kept invaders out of Arabia?
hot temperatures and deserts
cold temperatures and mountains
lack of water and cold temperatures
lack of water and mountains
The factors that kept invaders out of Arabia are hot temperatures and deserts. The Arabian Peninsula is predominantly covered by desert regions, including the vast Rub' al Khali or Empty Quarter, as well as the Syrian Desert and the Arabian Desert. These desert environments feature extreme heat, harsh aridity, and challenging terrain, making it difficult for invading forces to traverse the region and sustain themselves. The combination of inhospitable desert conditions and extreme temperatures acted as major deterrents for potential invaders, contributing to Arabia's historical isolation and relative protection from external threats.
Which of the following was the title of the Muslim empire's leaders just after Muhammad's death?
caliphs
priests
The title of the Muslim empire's leaders just after Muhammad's death was caliphs. Caliphs were the successors to Muhammad and served as the political and religious leaders of the Muslim community, known as the Ummah. They were responsible for governing the Muslim territories and upholding the teachings and principles of Islam.
Why did Islam spread through much of Southwest Asia during the 650s and 660s?
Muslim armies conquered the Sasanian empire.
Roman missionaries spread Islam to new regions.
The reason Islam spread through much of Southwest Asia during the 650s and 660s was that Muslim armies conquered the Sasanian Empire. The Sasanian Empire, which was the dominant power in the region at the time, was weakened by internal conflicts and exhaustion from its long wars against the Byzantine Empire. Taking advantage of this vulnerability, the Muslim armies led by the Rashidun and Umayyad caliphs launched successful military campaigns and swiftly conquered large parts of the Sasanian Empire, including its capital, Ctesiphon. These military victories facilitated the spread of Islam across the conquered territories, as the Muslim conquerors often offered favorable treatment and tolerated religious freedom for non-Muslim communities. This led to the assimilation and conversion of many people to Islam, solidifying its presence and influence in Southwest Asia.
Which form of literature was important to early Arab culture?
dramatic plays about Muhammad and his followers
oral poetry about their tribes and their enemies
The form of literature that was important to early Arab culture was oral poetry about their tribes and their enemies. Oral poetry played a significant role in Arab culture before the advent of written literature. It was a means of transmitting historical and cultural knowledge, preserving the memory of tribal traditions, and providing entertainment. Arab poets played a central role in society, composing and reciting poetry that celebrated heroic figures, commemorated important events, and expressed the values and ideals of their tribes. This rich tradition of oral poetry was highly valued and helped shape early Arab cultural identity.
Which contributed to the growth of trade in the Arab Muslim empire?
a long peace with Rome
a strong central government
Both factors contributed to the growth of trade in the Arab Muslim empire:
1. A long peace with Rome: The Arab Muslim empire established peaceful relations with the Byzantine Empire (successor to the Roman Empire) after the conquests of territories previously under Byzantine control. This long period of peace, known as the Pax Islamica, facilitated trade and economic interactions between the Arab Muslim empire and Byzantine territories. The stability provided by this peace allowed for the development of extensive trade networks and the flourishing of commerce across the regions.
2. A strong central government: The Arab Muslim empire had a centralized government that promoted trade and commerce. The rulers, known as caliphs, established a legal system, including Islamic commercial laws and regulations, that protected traders and facilitated trade transactions. They also invested in infrastructure, such as roads, bridges, and ports, to improve transportation and facilitate trade routes. The strong central government's support of trade helped create a conducive environment for economic growth and prosperity within the empire.