All Answers 358925 Answers

Volume = Pi x r^2 x h

S. Area  =[2 x pi x r x h] + [2 x pi x r^2]

48pi      =[2 x 3.141592 x 4 x h] + [2 x 3.141592 x 4^2], solve for h(height)

h            =2 feet

Volume =3.141592 x 4^2 x 2

               =100.53 feet^3

thanks for explaining in details! 

3)

Simplify the following:
(3^(1/3) + 24^(1/3) + 192^(1/3))^3

24^(1/3) = (2^3×3)^(1/3) = 2 3^(1/3):
(3^(1/3) + 2 3^(1/3) + 192^(1/3))^3

192^(1/3) = (2^6×3)^(1/3) = 2^2 3^(1/3):
(3^(1/3) + 2 3^(1/3) + 2^2 3^(1/3))^3

2^2 = 4:
(3^(1/3) + 2 3^(1/3) + 4 3^(1/3))^3

Add like terms. 3^(1/3) + 2 3^(1/3) + 4 3^(1/3) = 7 3^(1/3):
(7 3^(1/3))^3

Multiply each exponent in 7 3^(1/3) by 3:
7^3×3^(3/3)

3/3 = 1:
7^3×3

7^3 = 7×7^2:
7×7^2×3

7^2 = 49:
7×49×3

7×49 = 343:
343×3

343×3 = 1029:
 
 =1029

2)

Since 98 = 2 x 7^2, it, therefore, follows that if you multiply 98 by: 2^3 x 7^2 =392 you will have:

(98 x 392)^1/4 =14. Therefore k=392

1) 

Sqrt(7) x Sqrt(7 + 6) x sqrt(91) x sqrt(91 + 6) =

Simplify the following:
sqrt(7) sqrt(7 + 6) sqrt(91) sqrt(91 + 6)

91 + 6 = 97:
sqrt(7) sqrt(7 + 6) sqrt(91) sqrt(97 ) 

7 + 6 = 13:
sqrt(7) sqrt(13 ) sqrt(91) sqrt(97)

sqrt(7) sqrt(13) sqrt(91) sqrt(97) = sqrt(7×13×91×97):
sqrt(7×13×91×97)

7×13 = 91:
sqrt(91×91×97)

91×91 = 8281:
sqrt(8281×97)

8281×97 = 803257:
 sqrt(803257 ) 

sqrt(803257) = sqrt(91^2×97) = 91 sqrt(97):

 =91 sqrt(97)=896.246

If f(x)=x3, what is the value of f-1(8)/(f(8))-1?

Answered here:

Darren has borrowed 100 clams from Ethan at a 10% simple daily interest. Meanwhile, Fergie has borrowed 150 clams from Gertie at a 5% simple daily interest. In how many days will Darren and Fergie owe the same amounts, assuming that they will not make any repayments in that time period?

After n days Darren will owe   100+(10% of 100)*n   =  100+10n  clams

After n days  Fergie will owe    150+(5% of 150)*n     = 150+ 7.5n  clams

They will owe the same when these are equal

100+10n = 150+7.5n

2.5n         =  50

        n      =   50/2.5

        n      =   200/10

        n      =   20 days

Thanks Guest :)

Seems like there are two:

etc!

I'll leave you to prove it.

.

"The number of toy cars that Ray has is a multiple of 6. When he loses two of them, the number of cars that he has left is a multiple of n. If n is a positive even integer less than 10, then how many possible values are there for n?"

A multiple of 6 means the original number is divisible by both 2 and 3.  Losing 2 means what is left is also divisible by 2, but not by 3.  If it isn't divisible by 3 then it can't be divisible by 6.  Hence n can only be 2, 4 or 8.

Perhaps this will help:

Note the number of repetitions every 360°.

Thank you soooooooo much! Your solution made so much sense! 

So Angela's amount would be 8000*(1.06^20), and bob's would be 10000*0.07*20+10000. We can use a calculator to find the values (because seriously, who wants to calculate 1.06^20?), and then we find the absolute differences. 

I hope this helped!

expand  (3/5 x - y/2)^8

(6561 x^8)/390625 - (8748 x^7 y)/78125 + (5103 x^6 y^2)/15625 - (1701 x^5 y^3)/3125 + (567 x^4 y^4)/1000 - (189 x^3 y^5)/500 + (63 x^2 y^6)/400 - (3 x y^7)/80 + y^8/256 (9 terms).

It is the 7th term =8 nCr 6 * (3/5 x)^(8 - 7 + 1) * (-y/2)^(7 - 1)

Or: 63/400x^2y^6

Try this:

P = 4 and Q =4/2 =2, and b =P+2 = 4 + 2 =6. So you have:

P = 4

Q = 2

b = 6. But P(6) x P(6) =QP(6) =4(6) x 4(6) =24(6)

There are 3 solutions for x as follows:

x = 0, x = -1, and x = 1

Are you taking AoPS classes?

The side UD is 13 and from point D to the right angle symbol (let's say that point is A) is 5. From the pythagorean theorm, we know that line UA is 12. Now draw a perpendicular line from C to line DK, and let's call the point in which the line intercepts DK as B. Because UABC is a rectangle, we know CA is 12. From the pythagorean theorm again, we know that line AK=16. Now we know the base and height of triangle CBK. The base is 16, the height is 12, so the area is 96. From before, we also know the base and height of triangle UDA, in which 5 times 12 divided by 2 is 30. Finally, the rectangle in the center is 15 times 12 equals 180, and when you add them all up, you get 96+30+180=306. 

Check work: The height is 12, the base is 5+15+16=36, the top is 15, so from the trapezoid area formula, you get (15+36) times 12 divided by 2, which is 6 times 51 divided by 2, which is, again, 306.

@guest you were right too :)

what is the question?

The degree of a polynomial is determined by the largest power of x that occurs.  Multiplying by a constant, no matter how big, makes no difference. 

So, in "Let f(x) = x^4-3x^2 + 2 and g(x) = 2x^4 - 6x^2 + 2x -1. Let a be a constant. What is the largest possible degree of f(x) + a.g(x)?", the degree is 4.

Now, with that knowledge, see if you can do the other parts of your question.

My question keeps getting cut off. The last part is 

m >= 2 and a

Thanks! I tend to get really confused on these types of problems, so your solution helped me!

The nickel an the quarter doesn't affect it. We list out the possibilities.

p:h d:h

p:h d:t

p:t d:h

p:t d:t

2 out of the 4 combinations work, so the probability is 1/2.