# Category: 10th Grade

Friday, October 1st, 2010

### Solution:

- In his seventh year, the professor receives $70,000. Since the question does not state when he receives this amount, let’s assume it is received at the end of the seventh year as a lump sum. The present value of this is $46,554 (i.e. $70,000/(1.06^7))
- Similarly, the professor receives $70,000 at the end of year 14, and the present value of this is $30,961.07
- Similarly, the professor receives $70,000 at the end of year 21, and the present value of this is $20,590.88
- Similarly, the professor receives $70,000 at the end of year 28, and the present value of this is $13,694.11
- Similarly, the professor receives $70,000 at the end of year 35, and the present value of this is $9,107.37
- Similarly, the professor receives $70,000 at the end of year 42, and the present value of this is $6,056.92

Now we can add up each of the six present values, to arrive at the value of all the amounts received during the sabbaticals: $126,964.34 (if you add up the numbers above you get $126,964.35, but that’s due to rounding error).

If the professor receives the money at the middle of each year, you can re-calculate the above (e.g. by discounting the seventh year by 6.5 years rather than by 7 years). You might also want to recalculate this assuming that the salary is paid monthly, and see how it compares to the above result.

Saturday, September 4th, 2010

### Solution:

- The formula for the volume of a sphere is
- r = 6
- =

Saturday, September 4th, 2010

If you have a function called ƒ, let’s call its inverse ƒ^{–1} (i.e. ƒ^{–1} is the inverse function of ƒ).

By definition, the property of ƒ^{–1} is that if ƒ(a)=b, then ƒ^{–1}(b)=a. Wikipedia (http://en.wikipedia.org/wiki/Inverse_function) has a good example:

A function ƒ and its inverse ƒ^{–1}. Because ƒ maps a to 3, the inverse ƒ^{–1 } maps 3 back to a. In mathematics, if ƒ is a function from a set A to a set B, then an inverse function for ƒ is a function from B to A, with the property that a round trip from A to B to A (or from B to A to B) returns each element of the initial set to itself.

Friday, July 23rd, 2010

### Solution:

First write down the two equations and then pick several values of x and use those values to calculate the corresponding values of y. You can use the x,y pairs to then plot each line. You will see that the two lines intersect at 2,8 =i.e. x=2 and y=8).

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:

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: a^{2}=b^{2}+c^{2}– 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.

Thursday, July 22nd, 2010

### Solution:

Here’s how you solve this type of question:

- 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!
- One option is to buy no birds at all, and keep the $250.
- Since chickens cost only $1 each, he could buy 200 chickens and have $50 left over.
- Or he could buy 125 ducklings and have no money left over.
- 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.

Saturday, July 17th, 2010

### Solution:

1. Let’s call the number of cookies Cara made “x”

2. First she sold three fifths of x(.6x)

3. The remaining amount is two fifths of x(.4x)

4. Then she one fourth of x;two fifths of “x” (.25*.4x = .1x)

5. .6x=.1x+200

6. .5x=200

7. x= 400

Saturday, July 17th, 2010

### Solution:

A good absorber of radiant energy, like a black surface, sends to get hot. If it is a poor emitter, that means it will remain hot for a long time. Therefore, an object that is both a good absorber of radiant energy and a poor emitter would tend to be hotter than its surroundings.