I need to know how to do problems like this, I want to know how to solve, and simplify them, and also finding slopes. Like these kinds of problems(Write the equation, 3x + 5y = 15, in the slope intercept form (y = mx + b))

Solution:

1. Start with an equation like 3x + 5y = 15
2. Move 3x to the other side, so 5y = 15 – 3x
3. Divide both sides by 5, so y = 3 – x
4. Another way of writing this is y = -.6x + 3, or y=mx = b, where m=-0.6 and b = 3
5. If m = -0.6, that means that the slope = -0.6

Find the descriminant of x^2-7x+2=0 and describe the natural roots

Solution:

1. When you have an equation in the form of ax² + =x + c = 0, the solution is x = where  D = b² – 4ac
2. D stands for discriminant, and values of x are the natural roots

• If D > 0, then the values of x are real and distinct.

• If D = 0, then the values of x are real and equal

• If D < 0, then the values of x are unreal

1. In your question, x² – 7x + 2 = 0, so the values of a, b and c are as follows:

• a = 1

• b = -7

• c = 2

1. D = b² – 4ac = 49 – 8 = 41 (this is the value of the discriminant)
2. Since D > 0, then the values of x (the natural roots) are real and distinct

The Trail of Tears in 1838-1839 & the Battle of Wounded Knee in 1890 were significant because they A. demonstrate the technological superiority of the US troops over the indigenous peoples of North America B. typify the harsh treatment of Native peoples at the hands of the US government C. exemplify the determination of the American government to bring unity to North America & undermine any attempts at recession D. clarify the US government’s position on slavery & free soil” E. added large portions of land to the US which had been controlled by foreign powers

Solution:

We feel that the best answer is “B”, as both events involved the deaths of large number of native Americans, including non-combatants:

1. The Trail of Tears: “The Trail of Tears was the relocation and movement of Native Americans, including many members of the Cherokee, Creek, Seminole, and Choctaw nations among others in the United States, from their homelands to Indian territory (present day Oklahoma) in the Western United States. The phrase originated from a description of the removal of the Choctaw Nation in 1831. Many Native Americans suffered from exposure, disease, and starvation while en route to their destinations, and many died, including 4,000 of the 15,000 relocated Cherokee.” (Wikipedia)
2. Battle of Wounded knee: “On December 29, 1890, 365 troops of the U.S. 7th Cavalry regiment, supported by four Hotchkiss guns, surrounded an encampment of Miniconjou (Lakota) and Hunkpapa Sioux (Lakota) near Wounded Knee Creek, South Dakota. The Sioux had been cornered and agreed to turn themselves in at the Pine  ridge Agency in South Dakota. They were the very last of the Sioux to do so. They were met by the 7th Cavalry, who intended to disarm them and ensure their compliance. During the process of disarming the Sioux, a deaf tribesman named Black Coyote could not hear the order to give up his rifle and was reluctant to do so. A scuffle over Black Coyote’s rifle escalated into an all-out battle, with those few Sioux warriors who still had weapons shooting at the 7th Cavalry, and the 7th Cavalry opening fire indiscriminately from all sides, killing men, women, and children, as well as some of their own fellow troopers. The 7th Cavalry quickly suppressed the Sioux fire, and the surviving  Sioux fled, but U.S. cavalrymen pursued and killed many who were unarmed. By the time it was over, about 146 men, women, and children of the Lakota Sioux had been killed. Twenty-five troopers also died, some believed to have been the victims of friendly fire as the shooting took place at point blank range in chaotic conditions. Around 150 Lakota are believed to have fled the chaos.” (Wikipedia)

In trapezoid ABCD the median MN cuts diagonals AC =D at p and q repectively, BC=12, and AD=20 Determine PQ please explain your steps. i know the median is 16 and I know the correct answer is 4, I know there is a formula but we have not proved it.

Solution:

First, let’s draw the trapezoid so that we can visualize the problem. We have not drawn the trapezoid exactly to scale, but this should be accurate enough for our purposes:

Now, we know that MN = 16 because the median equals the average of the top and bottom of the trapezoid

We also know that the distance of Mp = 6, because it just be half the length of BC (the sides of triangle AMp are half the size of the sides of triangle ABC because the triangles are similar and we know that AM is half the length of AB since MN is the median).Similarly,the length of qN = 6.

Now it’s easy to solve for pq:

1. Mp + pq + qN = MN = 16
2. 6 + pq + 6 = 16
3. pq = 4

If i was just given the container specification which is the shape of base is triangle, the height of box 40 in and the volume 720 cu in. the question is what dimensions were necessary for you to deteremine before you could build your box. How did you find the missing dimensions?

Solution:

Let’s assume that the sides of the box are vertical. In that case, the volume of the container would be the area of the base times the height. Since we know the volume is 720 cu. in., and the height is 40 =n, then we know that the area of the base is 18 sq. in. = 18*40=720).

We don’t have enough information to find the lengths of the sides of the container’s triangular base. There are many shapes of triangles whose area is equal to 18 sq. inches. We would need to know the lengths of at least two sides of the base of the container, plus the angle formed by these two sides. We can then use the cosine rule (see below) to figure out the required length of the third side.

The “cosine rule”

In the triangle below, the three sides have lengths a, b and c. Angle A is opposite side a; similarly for B and C.

The cosine rule is as follows:  a²=b²+c²– 2bc (cos(A))

This allows us to work out the required length of the third side of the triangle if we know the length of two sides and the angle between them.

How do you find the area of a base on a 3-D rectangle?

Solution:

1. The base of the 3D rectangle is in the shape of a 2D rectangle.
2. Multiply the length of this 2D rectangle by its width to let the area

You are developing new bath soap, and you hire a public opinion survey group to do some market research for you. The group claims that in its survey of 450 consumers, the following were named as important factors in purchasing bath soap: Odor 425 Lathering Case 397 Natural Ingredients 340 Odor and lathering Case 284 Odor and Natural Ingredients 315 Lathering ease and natural ingredients 219 All three factors 147 should you have confidence in these results? Why or why not?

Solution:

First, take a look at this link to understand the Principle of Inclusion and Exclusion:

http://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle

Next, make a list of all the subsets:

Has to do with triangle, saw of cosine and sin, etc. I have to draw a polygon with 6-10 sides, and provide enough info for someone to be required to find 3 or more missing angles and 3 or more missing sides. Solve and find the area of the polygon.

Solution:

We’ll try the case of a six-sided polygon and will show that if we provide 3 angles, 3 sides and 5 spokes, we can expect someone to figure out the size of the three unknown angles and the lengths of the three unknown sides.

Here’s an example of a six-sided polygon, an irregular hexagon:

As you can see, this is an irregular shape – it’s not a regular hexagon where the sides and angles are of the same size.

Let’s now draw a central point and spokes, where you know the length of each spoke and all but one of the angles between the spokes:

You can calculate the unknown angle (we can call it “F”) because we know that all the angles need to add up to 360° (F = 360 – A – B – C – D – E). Since we know the length of two sides and one angle of each triangle that is formed (i.e. side-angle-side, or SAS for short), one can calculate the unknown side of each triangle using the Law of Cosines. Are you familiar with the Law of cosines and how it applies to an SAS case? You can find a lot of explanations online, but we can help you with this if you’re not able to find anything that you find easy to understand.

Now, let’s assume we know the length of three of the sides. If we know the length of the spokes and three of the sides (a, b and d shown below), then we only need to know two of the angles between the spokes (since the angles A, B, and D can be calculated using the Law of cosines) :

Now, let’s assume that we’re given two of the angles between the sides of the polyhedron (X and Y, shown below). In this case, we don’t need any of the angles between the spokes (E and C can be calculated using the Law of Sines, since this is a side-side-angle case, or SSA for short):

If we know one more angle (Z,  shown below), then we can give up having to know one of the spokes, since we can calculate the length of the “unknown spoke” – shown with a dashed line – using the Law of Cosines (which applies because this is a side-angle-side case, or SAS for short):

As you can see from the above diagram, if we provide three angles (X, Y and Z) and three sides (a, b and c) and 5 spokes of an irregular hexagon, we can expect someone to figure out the size of the unknown angles and the lengths of the unknown sides.

There are many ways to tackle this type of problem . we’ve shown another example below.

Here’s another approach.

First, let’s provide some information on some of the sides and angles:

As you can see, we have indicated the lengths of three out of the six sides, and three out of the six angles (the diagram is not drawn to scale).

Now let’s draw a couple of lines as follows:

We can find the length of the blue line opposite the 95° angle by recognizing that we know the length of two sides and the angle between them (side-angle-side, or SAS for short). To find using the unknown side, we =an use cosine rule a² = b² + c² – 2bc cos(A), where A is the angle that we know (in this case 95°).

So we can re-write a² = b² + c² – 2bc cos(A) as follows:

a² = 9 + 4 – 12 cos(95°) = 13 – 12*(-0.087155) =13+ 1.0458689 = 14. 0458689, so a = 3.75.

We can now figure out the angle B using the fact that =² = a² + c² – 2ac cos(B): 9=  14. 0458689 + 4 – 2*3.75*2*cos(B).

We can now solve for cos(B) =  0 .6031

Using a lookup table or a scientific calculator we can see that B = 52.39.

Since we know that the angles of a triangle add up to 180°, we know A+B+C=180°. We know that A = 95 and B = 52.39, so C must equal =2.61. Now that we know that shape and size of the triangle, we can calculate its area. The area equals half the base times the height. The base is 3.75 and we can calculate the height by taking the sin of B and multiplying it by c =height =sin(0.6031)*2 = 0.792183*2 = 1.5844). Therefore the area of this triangle (with angles A, B and C) is 0.5*3.75*1.5844 = 2.9709.

We can now tackle the sides, angles and area of the next triangle, where one of the sides is equal to 2.5, and the other is 3.75:

We know that the angle between these two sides is equal to 95.61, since B = 52.39 and the sum of the two angles equals 148.

So now we have a similar problem, where we know the length of two sides and the size of the angle that they form (side-angle-side, or SAS). We can solve this just like we solved the problem above.

Once we have solved this second triangle, we’re Left with a quadrilateral for which we know only one side (d) and one angle (X):

There are too many unknowns to solve this, so we’ll need to provide a little more information so that someone can find length of the remaining sides and the size of the remaining angles.

One way to solve this is to break up the remaining area into smaller triangles, and provide information on some of the segments and angles:

As you can see from the above, we have created four new triangles, three of which are right angles. If we know p, q and r and the angle P, we can calculate the size and shape of each of the four triangles.

How do you find the derivative to the function d(x)= 3x^2 e^x + 2xe^2 + 3e^x ?

Solution:

A farmer has enough space in his farm to rear 200 birds. He buys chickens at $1 each and ducklings at $2 each. He cannot spend more than $250 for purchasing the birds. What are the possible numbers of birds he can purchase?

Solution:

Here’s how you solve this type of question:

1. Assume that the maximum number of birds he can purchase’s 200, since he does not have space for more than that. Of course, he could purchase more and then release them into the wild, but we don’t think that’s what your teacher has in mind!
2. One option is to buy no birds at all, and keep the $250.
3. Since chickens cost only $1 each, he could buy 200 chickens and have $50 left over.
4. Or he could buy 125 ducklings and have no money left over.
5. So there are lots of possibilities. The general formula would be given by c + 2d 250 and c + d  200, where c is the number of chickens and d is the number of ducklings. Since we are dealing with a real world situation, we can also assume that c and d are non-negative integers. The solution space would look like this:

The above chart shows all the possible numbers of chickens and ducklings that would satisfy the constraints of space and money. As you can see, there are thousands of options.