All Answers 358407 Answers

Yes, kevbamboo, I well understand that there are multiple ways of approaching this problem. 

Your way completely skips steps a, b, c, and d. 

If you want to get technical, I can use 4 different methods to calculate the sum, and start my solution off with "this is easy". 

However, the method the questioner requests is obviously the one presented in my solution. 

Steps a, b, c, and d are just parts of the proof and investigation of the method. 

This is easy. The formula for the sum of a set of consecutive numbers is, when starting from one, (1+n)n/2. Here n equals 200. 200*201/2=100*201=20100

\(1+2+3+\cdots+198+199+200=?\)

a. 

First number: 1, 

Last number 200. 

\(1+200=\boxed{201}\)

b. 

Second number: 2, 

Second to last number: 199.

\(2+199=\boxed{201}\)

c. 

Third number: 3, 

Third to last number: 198.

\(3+198=\boxed{201}\)

d. 

When you pair up all the numbers as suggested above, each pair adds to 201. 

Since there are 2 numbers in each pair, and there are 200 numbers, there are 200 ÷ 2 = 100 pairs. 

e. 

Since each pair sums to 201, and there are 100 pairs, \(1+2+3+\cdots+198+199+200=201\cdot100=\boxed{20100}\)

I'd like to see someone answer the second question too :)

---

I do not think I can help with the second question. 

And I do not even understand what you are asking with the first question. (That may or may not be because of my own ignorance)

Given that x is an integer such thatx * √x - 5x - 9 * √x = 35 , find x.

\(x\sqrt x - 5x - 9\sqrt x = 35\qquad x\ge 0\\ let\;\;y=\sqrt x\qquad \qquad \qquad y\ge 0\\ y^2*y-5y^2-9y=35 \\ y^3-5y^2-9y=35 \\ y^3-5y^2-9y-35 =0\\ \)

I am told that there is an integer solution to this so how do I find it ?

Well I know y must be positive, that makes it easier.

The solution must be a factor of 35.

So it has to be 1 or 5 or 7 or 35.

1-5-9-35 = -48    so 1 is not right

5^3-5*5^2-9*5-35 = -80 so 5 is not correct

7^3-5*7^2-9*7-35 = 0    so 7 is a root   

35^3-5*35^2-9*35-35 = 36400 so 35 is not a root.

So the only poisitive integer solution is y=7

so

\( x=y^2 \\x= 49\)

-------------

Graphical check :

When I graphed  \(y^3-5y^2-9y-35=0\)   

this is the the result ... just remember that the axis have been reversed.

I know this is a cubic so the cruve cannot turn around again so the only answer is y=7

Which means that x=49.

"Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a diamond and the second card is an ace?"

.

The shaded region shown consists of 11 unit squares and rests along the $x$-axis and the $y$-axis. The shaded region is rotated about the $x$-axis to form a solid. In cubic units, what is the volume of the resulting solid? Express your answer in simplest form in terms of $\pi$.

A large sphere has a volume of $288\pi$ cubic units. A smaller sphere has a volume which is $12.5\%$ of the volume of the larger sphere. What is the ratio of the radius of the smaller sphere to the radius of the larger sphere? Express your answer as a common fraction.

Hi guest, 

Your answer only include pizzas with all four toppings.

You have to inlude smaller numbers of toppings as well :)

Hi Gavin  and others,

When people make a post like this I think it is intended as a game.

The asker is implying that he/she knows the answer and is asking you to accept the private challenge to answer it.

I personally would be unlikely to answer a question like this at all, I do not think it is intended for the mathematicians that frequent this site regularly.

Then again, with a heading like 'help' I am quite likely wrong, and I would advise people to simply not answer at all.

No, the answerer cannot get points for a private answer.

I do take your point that it stops others learning from the answer but if lol is a nice person I would expect him/her to be prepared to provide the answer privately.

Perhaps lol is hoping to attract like minded people to answer him privately so that he/she can go on and form a private freindship with one or more of them? 

See the answer here: https://web2.0calc.com/questions/geometry-prisms-question

The solution to this is quite involved:
Solve for x:
-9 sqrt(x) - 5 x + x^(3/2) = 35

Subtract 35 from both sides:
-35 - 9 sqrt(x) - 5 x + x^(3/2) = 0

Simplify and substitute y = sqrt(x).
-35 - 9 sqrt(x) - 5 x + x^(3/2) = -35 - 9 sqrt(x) - 5 (sqrt(x))^2 + (sqrt(x))^3
 = y^3 - 5 y^2 - 9 y - 35:
y^3 - 5 y^2 - 9 y - 35 = 0

The left hand side factors into a product with two terms:
(y - 7) (y^2 + 2 y + 5) = 0

Split into two equations:
y - 7 = 0 or y^2 + 2 y + 5 = 0

Add 7 to both sides:
y = 7 or y^2 + 2 y + 5 = 0

Substitute back for y = sqrt(x):
sqrt(x) = 7 or y^2 + 2 y + 5 = 0
Raise both sides to the power of two:
x = 49 or y^2 + 2 y + 5 = 0

Subtract 5 from both sides:
x = 49 or y^2 + 2 y = -5

Add 1 to both sides:
x = 49 or y^2 + 2 y + 1 = -4

Write the left hand side as a square:
x = 49 or (y + 1)^2 = -4

Take the square root of both sides:
x = 49 or y + 1 = 2 i or y + 1 = -2 i

Subtract 1 from both sides:
x = 49 or y = -1 + 2 i or y + 1 = -2 i
Substitute back for y = sqrt(x):
x = 49 or sqrt(x) = -1 + 2 i or y + 1 = -2 i

Raise both sides to the power of two:
x = 49 or x = -3 - 4 i or y + 1 = -2 i

Subtract 1 from both sides:
x = 49 or x = -3 - 4 i or y = -1 - 2 i

Substitute back for y = sqrt(x):
x = 49 or x = -3 - 4 i or sqrt(x) = -1 - 2 i

Raise both sides to the power of two:
x = 49 or x = -3 - 4 i or x = -3 + 4 i

-9 sqrt(x) - 5 x + x^(3/2) ⇒ -9 sqrt(-4 i - 3) - 5 (-3 - 4 i) + (-3 - 4 i)^(3/2) = -5 + 40 i:
So this solution is incorrect
-9 sqrt(x) - 5 x + x^(3/2) ⇒ -9 sqrt(4 i - 3) - 5 (-3 + 4 i) + (-3 + 4 i)^(3/2) = -5 - 40 i:
So this solution is incorrect

-9 sqrt(x) - 5 x + x^(3/2) ⇒ -9 sqrt(49) - 5 49 + 49^(3/2) = 35:
So this solution is correct

The solution is:
x = 49

The first pizza, you have 4 choices to choose from, the 2nd pizza you have 3 choices, the 3rd you have 2 choices, and the 4th just 1 choice. So, you can order:

4! =4 x 3 x 2 x 1 = 24 number of different pizzas you can order.

Simplify the following:
(-8 i + 32)/(3 i + 5)

Factor 8 out of 32 - 8 i giving 8 (4 - i):
(8 (-i + 4))/(3 i + 5)

Multiply numerator and denominator of (8 (-i + 4))/(3 i + 5) by 5 - 3 i:
(8 (-i + 4) (-3 i + 5))/((3 i + 5) (-3 i + 5))

(5 + 3 i) (5 - 3 i) = 5×5 + 5 (-3 i) + 3 i×5 + 3 i (-3 i) = 25 - 15 i + 15 i + 9 = 34:
(8 (-i + 4) (-3 i + 5))/34

The gcd of 8 and 34 is 2, so (8 (-i + 4) (-3 i + 5))/34 = ((2×4) (-i + 4) (-3 i + 5))/(2×17) = 2/2×(4 (-i + 4) (-3 i + 5))/17 = (4 (-i + 4) (-3 i + 5))/17:
(4 (-i + 4) (-3 i + 5))/17

(4 - i) (5 - 3 i) = 4×5 + 4 (-3 i) - i×5 - i (-3 i) = 20 - 12 i - 5 i - 3 = 17 - 17 i:
(4 -17 i + 17)/17

Factor 17 out of 17 - 17 i giving 17 (1 - i):
(4×17 (-i + 1))/17
Combine powers. (4×17 (-i + 1))/17 = 4×17^(1 - 1) (-i + 1):
4×17^(1 - 1) (-i + 1)

1 - 1 = 0:
4×17^0 (-i + 1)

17^0 = 1:
4 (-i + 1) = 4 - 4i - Ramanujan's number.

Multiply the hours that both of you worked x amount of money you earned per hour. Then equate them and solve for t as follows:

 (t - 6) * (2t - 5) = (2t - 8) * (t - 5), solve for t

(t - 6) (2 t - 5) = (t - 5) (2 t - 8)

Expand out terms of the left hand side:
2 t^2 - 17 t + 30 = (t - 5) (2 t - 8)

Expand out terms of the right hand side:
2 t^2 - 17 t + 30 = 2 t^2 - 18 t + 40

Subtract 2 t^2 - 18 t + 40 from both sides:
t - 10 = 0

Add 10 to both sides:
t = 10 

If I understand your question, you have a right-angle triangle with one side=3 and the hypotenuse =7.

So, by Pythagoras's Theorem:

7^2 - 3^2 =T^2, T=Third side

T =6.325 miles.

Area of this triangle = 6.325 x 3 / 2 =~9.487 miles^2

Since the property is a square, then:

Total area of Jonah's property=9.487 x 2 =~18.974 miles^2.

ab - 3a + 4b = 137

a = (137 - 4 b)/(b - 3), or:
b = (3 a + 137)/(a + 4)
For positive solutions, I see only 3 as follows:
a =1 and b=28
a=21 and b=8
a=121 and b=4
From the 3 above, the one with smallest absolute difference is:
abs[21 - 8] = 13

any help?

See solution No. 4 here. change your number from 720 yards to 800 yards.

https://web2.0calc.com/questions/the-width-of-the-river

Never mind, I figured it out!
Thanks you all so much.

Never mind, I figured it out!
Thanks you all so much.

ab =1/2[10a + b], solve for a, b

a = b/(2 (b - 5)), or:

b = (10 a)/(2 a - 1)

If you try the first few numbers, you will quickly see that:

a =3  and b = 6. So, the number is:

36. Because: 3 x 6 =18, which is 1/2 of 36.