Some tricky problems

[Hockeyman]

Recieved these tricky problems for homework and i'm not sure how to do some of them.

1) what is the probablity that the roots of ax^2+bx+c= 0 are all real numbers if a,b,c are randomly cosen positve integers?

2) If a is 35 percent less than B and if C is 75 percent more than B, what percent of C is A

3) Find all positve integral values of n for which n^4 + 4 is a prime number

4) A circle of radius 3 intersects a circle of radius 5. Find the difference in the areas of the nonoverlapping regions.

If you could just point me in the right direction it would be greaty appreciated. Thank you

 

[sgtpepper]

4.) The area of the bigger circle minus the area of the smaller circle should be your answer, unless the circles intersect in some different way.

click here for link to calculation

 

[sgtpepper]

2) say that B is 100, then A is 65 and C is 175
so the question (2) is asking what percent of C is A

so A of C times 100 for percent
so A/C times 100 for percent
so 65/175 times 100 for percent
so 0.37 times 100 for percent
so 0.37 * 100
=
37%

http://instacalc.com/?d=&c=KDY1LzE3NSl8 ... s=ss&v=0.9

unless I misread/interpreted

 

[morson]

1) The roots are real if \(\displaystyle \L\ b^2 > 4ac\) or if \(\displaystyle \L\ b^2 = 4ac\). If I picked an 'a' value from a hat of positive integers, and did the same for 'b' and 'c', I'd be choosing from a set of {1, 2, 3, 4, 5, 6, 7 ... } There are an infinite number of elements in the hat. You can be sure that b being one is a certain failure, and b being 2 would likely be a failure, unless a and c are both 1. You should probably do a few trials and see what happens. I don't know if there is even a definite chance of an event like this occurring.

2 and 3 were covered above.

4) Let's call the area of the larger circle, excluding the overlapping region, A. Call the area of the smaller circle, excluding the overlapping region, B. Call the area of the overlapping region r. We know that A + r is the overall area of the larger circle, which is \(\displaystyle \L\ 25\pi\\) and B + r is the overall area of the smaller circle, which is \(\displaystyle \L\ 9\pi\\).

So, \(\displaystyle A + r = 25\pi\ - [1]\)
\(\displaystyle B + r = 9\pi\ - [2]\)

Subtract [2] from [1].

 

[soroban]

Hello, Hockeyman!

2) If A is 35 percent less than B and if C is 75 percent more than B,
what percent of C is A?

"A is 35% less than B" \(\displaystyle \;\Rightarrow\;\;A \,=\,0.65B\)

"C is 75% more than B" \(\displaystyle \;\Rightarrow\;\;C \,=\,1.75B\)

Therefore: \(\displaystyle \L\:\frac{A}{C} \:=\:\frac{0.65B}{1.75B} \:=\:\frac{13}{35} \:=\:37\frac{1}{7}\%\)