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3. How many factors of 1200 can be divided by 15 without a remainder?

1200  factors as 

120 * 10  =

15 * 8 * 2 * 5 =

15 * 2^3 * 2 * 5  = 

15  * 2^4 * 5

So....the number of factors of    2^4 *  5  =  (4 + 1) (1 + 1) = 5 * 2  = 10

So

10 factors of 1200 are divisible by 15

tenemos un foro en español

 https://web2.0calc.es/

CPhill: You have a reversal typo in your last answer. I wrote a short computer code to count them and it agrees with yours(corrected).

Rom: Yours is to be subtracted from 900,000 to get the right number.

def getCount(x) : count = 0; for i in range(100000, 999999) : count = count + has7(i); return count; print("Total = ", getCount(x)): Total =427,608 numbers that have at least one 7 in them.

2. What two-digit number has exactly 9 factors? 

The number is

2^2 * 3^2  =  36

1. How many non-empty subsets containing only prime numbers does the set {1, 2, 3, 4, 5, 6, 7} have?

We have four primes

The number of subsets formed by choosing any prime  = C(4, 1)  = 4

The number of subsets formed by choosing any two of the primes = C(4, 2) = 6

The number of subsets formed by choosing any 3 of the primes  = C(4,3) = 4

And...if we are allowed an improper subet.....one more containing all the primes

So  

4 + 6 + 4 + 1  =  15 subsets

Note that this is just the sum of the elements of the 4th row of Pascal's Triangle - 1  =

2^4 - 1    =   15

4.   All eight vertices of a unit cube are on a sphere (i.e. the cube is inscribed in the sphere). What is the surface area of the sphere?

The long diagonal of the cube will  = sqrt (3) units

The radius of the sphere is 1/2 of this

So....the surface area of the sphere =

4 pi [ radius]^2  =  

4 pi [ sqrt (3) / 2 ] ^2  =

4 pi (3/4)   =

3 pi units^2

3.   The surface area of a sphere is 36pi. Find the volume of the sphere.

SAsphere  =    4 pi ^ r^2

36 pi  =  4 pi * r^2

36 = 4r^2

9 = r^2

3 = r

So.....

Vsphere  =   (4/3) pi * r^3   =  (4/3) pi (3)^3  =  4 pi (3)^2  = 36 pi units^3

2.   A plane cuts through a sphere with diameter 20 cm, but the closest it gets to the center is 3 cm. What is the area of the intersection of the sphere and the plane in sq cm?

Looking at a cross-section of thec sphere....we have a great circle with a dadius of 10 units

Let the equation of this circle be x^2 + y^2 = 100

The intersection of the plane and sphere will create a smaller circle

Let y = 3  and we can solve for x^2  =  the radius of the smaller circle

x^2 + (3)^2  =  100

x^2  = 100 - 9

x^2 = 91 = r^2

So....the area of this smaller circle  =  pi * r^2   =    91 pi cm^2

yo hablo espanol

Integers that contain no 7s  =

(8 choices) for 1st digit    [ not 0 or 7 ]

(9 choices) for each of the next 5 digits  [ no 7s ]   = (9)^5

So....

8 * 9^5  =  472392 

And we have  999,999 - 100,000 + 1  = 900,000 possible 6 digit integers

So.......the number that contain at least one 7  =

900,000 - 472392  = 

427608

\(\text{Let } z=a+bi\text{ and }w=c+di\\\text{Then }|z|=\sqrt{a^2+b^2}\text{, }|w|=\sqrt{c^2+d^2} \text{ and }z\bar w=(a+bi)(c-di)=ac+bd+(bc-ad)i\)

\(\text{Hence }a^2+b^2=5^2,ac+bd=6,c^2+d^2=2^2,bc-ad=8\)

(1)

  \(|z+w|=|a+c+(b+d)i|=\sqrt{a^2+c^2+2ac+b^2+d^2+2bd}=\sqrt{a^2+b^2+c^2+d^2+2(ac+bd)}\\=\sqrt{5^2+2^2+2\times6}=\sqrt{41}\\\text{So }|z+w|^2=41\)

See if you can do the rest using a similar approach.  

sqrt (18)  = sqrt (9 * 2)  =  sqrt 9  * sqrt 2  =  3 sqrt(2)

sqrt (24) =  sqrt (4 * 6)  =  sqrt 4 * sqrt 6   =  2sqrt(6)

Total area  = [ 3 sqrt (2) ] * [ 2 sqrt (6) ]  =  [ 6 *sqrt (2) * sqrt (6) ] = 6 sqrt (12)  =

6 sqrt ( 4 * 3)  =   6  * sqrt (4) * sqrt (3)  =  6 * 2 * sqrt (3)  =   12 sqrt (3) units^2

If we cut the length and width in half......the new area would be

(1/2) * 3 * sqrt(2)  *  (1/2) * 2 * sqrt (6)  =

(3/2) * sqrt (2) * sqrt (6)  =

(3/2) sqrt (12)  =

1.5 sqrt (4) * sqrt (3)  =

1.5 * 2 * sqrt (3)  =

3 sqrt (3)  units^2

Part A

Here's the way I partitioned this, Nickolas

We have two triangles and one rectangle

Part B  

AB = 4 units

AE = 5 units

Part C 

Area of rectangle ABGE = 4*5 = 20 uits^2

Area of triangle BGC = (1/2)base * height = (1/2)(3) (5) = 7.5 units^2

Area of triangle ECD  = (1/2)base * height = (1/2)(7)(2) = 7 units^2

So....total area =   20 + 7.5 + 7  =  34.5 units^2

Part A - I believe that net A is the correct one

Part B  -   AB  = 3 in     BC = 5 in   CD = 4 in

Part C -  Surface Area

We have two right triangles with leg of 3 and 4

So....their areas =  2(1/2)(product of leg lengths) =  3 * 4 =  12 in^2

Base  is a   4 * 8.6  rectangle =  34.4 in^2

Top is a  5 * 8.6 rectangle = 43 in^2

Back is a 3 * 8.6 rectangle = 25.8 in^2

So....the total surface area  =  12 + 8.6 (3 + 4 +  5)  = 12 + 8.6(12) = 12 (1 + 8.6) = 12 * 9.6   = 115.2 in^2

No it isn't wrong.

I just asked 2 other Native english speakers who said Browsing as well.. Maybe teacher got confused there :P Thanks :D 

thank you so much too!!!