Friday, July 23rd, 2010
Solution:
First, you have to have a clear idea what y = 2x^2+4x-5 means. This is an equation that tells you that, for any value of x, you can find y by dividing by multiplying x by x, and then multiplying that by two, adding 2 times x and finally subtracting 5. Does that sound familiar?
Now, what is the table you have to make? You will list several values of x (which you will be using as x-coordinates in your graph), and for each one use the equation to find what y is. We’ll show the work:
For x = 1, y = 2*1*1 + 4*1 – 5 = 1
For x = 2, y = 2*2*2 + 4*2 – 5 = 11
For x = 3, y = 2*3*3 + 4*3 – 5 = 25
For x = 4, y = 2*4*4 + 4*4 – 5 = 43
For x = 5, y = 2*5*5 + 4*5 – 5 = 65
Do you see what we did there? We just replaced x in the equation by each value of x, and did the multiplication and addition.
Now your table will look like this:
x | y
–+—–
1 | 1
2 | 11
3 | 25
4 | 43
5 | 65
We’ve only shown the x,y values for 1 through 5, out there are an infinite number of x values that you could plug in, including negative numbers and fractions. You can use the resulting T chart to help you create a graph, which will look like this:
Friday, July 23rd, 2010
Solution:
First, let’s define what a Newton (N) is: “The Newton (N) is defined as the amount of force that, when acting on a 1 kg mass, produces an acceleration of 1 m/s/s (one meter per second per second). Therefore, 1 N = 1 kg =D7 1 m/s/s.”
The force of friction opposing the 325 N force is .25 * 925 N (which is equal to 231.25 N). The force pulling the crate is 325 N * cos(25), i.e. 294.55 N. (This is the component of force acting parallel to the floor.)
Take the difference between these two and you will have the net force pulling the crate (which is 63.3 N). Using Newton’s equation F = ma, you will find the acceleration by dividing the net force by the mass of the crate (the mass is 925 =/(one standard gravity [9.80665 m/s/s], or 94.32 kg). The acceleration is therefore 63.3 N / 94.32 kg, which equals 0.671 m/s/s.
Friday, July 23rd, 2010
Solution:
The total area is 40π=2πrh+2πr^2
=2πr*8 + 2πr²
=16πr + 2πr²
=16r + 2r²
If we rearrange the terms and divide the terms by 2 (in order to simplify the equation), we get:
r2 + 8r – 20 = 0
(r+10) (r-2) = 0
r = 2 or -10
Therefore, r = 2 (assuming that -10 is not an option!).
Friday, July 23rd, 2010
Solution:
We feel that the best answer is “B”, as both events involved the deaths of large number of native Americans, including non-combatants:
- 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)
- 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)
Friday, July 23rd, 2010
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:
- Mp + pq + qN = MN = 16
- 6 + pq + 6 = 16
- pq = 4
Friday, July 23rd, 2010
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.
Friday, July 23rd, 2010
Solution:
1. The cardinality of a set is a measure of the “number of elements of the set”. For example, the set A = {2, 4, 6} contains = elements, and therefore A has a cardinality of 3.
2. Given a set S, the power set of S, written , P(S), is the set of all subsets of S, including the empty set and S itself.
3. If S={a, b}, then P(S) ={{},{a},{b},{a,b}} and the Cardinality of P(S) is 4.