All Answers

+1

...as an edit .... PERHAPS you could say since there is ONE minute more than five 5 minute periods, that ONE more bacterium was produced

(but the question does NOT say the bacteria reproduce 1 per minute....it says 5 every 5 minutes)

SO if you go with the one extra bacterium 137 - 26 = 111 originals .....

GuestFeb 14, 2021

#1

+1

506 minutes is 8 hrs 26 minutes

so the 'initial ones' have had 26 minutes to reproduce

26 minutes is five 5 minute periods so the increase is 5 x 5 = 25

137 - 25 = 'initial ones' = 112

GuestFeb 14, 2021

#3

+118703

+1

Yes that'sright.

The number in front of the y is doubled so the number in front of the x must be doubled too.

(So long as they are both are on the same side of the equal sign that is fine. )

y=11x+14

2y=ax+4

a=11*2=22

MelodyFeb 14, 2021

#1

+1

Solution by 3D vecors! The pyramid is spanned by u = (4,5,0), v = (4,0,6), and w = (0,5,6). Taking the determinant gives us a volume of 42.

GuestFeb 14, 2021

#1

+592

+2

Since we know that a*b = √(a + b) / √(a - b), all we need to do is replace x for a and 24 for b.

x*24 = √(x + 24) / √(x - 24) = 7

√(x + 24) = 7√(x - 24)
(x + 24) = 49(x - 24)
x + 24 = 49x - 1176
24 + 1176 = 49x - x
  1200 = 48x

x = 25

LogarhythmFeb 14, 2021

#3

+506

+1

Nice solution guest and logarhythm!

Here's an alternate solution:

We need to find the coefficients of x^2, x, and constants of both $(1+2x)^5$ and $(2-x)^6$ .

To do that, use the binomial theorem (explained here: https://www.mathsisfun.com/algebra/binomial-theorem.html )

Knowing that, the constant, x, and x^2 coefficients of $(1+2x)^5$ are 5 choose 0 = 1 , 5 choose 1 * 2 = 10, and 5 choose 2 * 2^2 = 40, respectively, and the constant, x, and x^2 coefficients of $(2-x)^6$ are 6 choose 0 * 2^6 = 64, 6 choose 1 * -1 * 2^5 = -192, and 6 choose 2 * 2^4 = 240, respectively.

The x^2 term of the product of the 2 expressions will happen when an x term of one expression is multiplied by the x term in the other expression, or when the x^2 term in one expression is multiplied by a constant term in the other expression.

That is equal to $64\cdot40+-192\cdot10+240\cdot1=\boxed{880}$

.

textotFeb 14, 2021

#2

0

Sorry I just realized there is a much quicker way to figure this out.

Going back to this step:

(1+2x)5(2-x)6 = (32x⁵ + 80x⁴ + 80x³ + 40x² + 10x + 1)(x⁶ - 12x⁵+ 60x⁴ - 160x³ + 240x² - 192x + 64)

You can find the coefficient of x² by multiplying the terms that make x² and adding them up:

(10x)(-192x) + (40x²)(64) + (1)(240x²)

= -1920x² + 2560x² + 240x²
= 880x²

Which means 880 is your answer :)

GuestFeb 14, 2021

#1

+592

+1

This isn't the best way to do it since it's just brute force but it's still an answer I guess:

I'm sure you've learned how to expand brackets right? If you haven't, you basically expand it like this:

Now, we can expand all of this:

(1+2x)⁵(2-x)⁶

--> (1+2x)⁵ = (1+2x)(1+2x)(1+2x)(1+2x)(1+2x)
= 32x⁵+ 80x⁴ + 80x³ + 40x² + 10x + 1

--> (2-x)⁶ = (2-x)(2-x)(2-x)(2-x)(2-x)(2-x)
= x⁶ - 12x⁵ + 60x⁴ - 160x³ + 240x² - 192x + 64

So if we put this together:

(1+2x)⁵(2-x)⁶ = (32x⁵+ 80x⁴ + 80x³ + 40x² + 10x + 1)(x⁶ - 12x⁵ + 60x⁴ - 160x³ + 240x² - 192x + 64)

Then when you expand this, you get:

32x¹¹ - 304x¹⁰ + 1040x⁹ - 1240x⁸ - 790x⁷ + 2537x⁶ + 76x⁵ - 2180x⁴ - 320x³ + 880x² + 448x + 64

But all you need to pay attention to is 880x² which means the coefficient of x², in other words your answer, is 880.

LogarhythmFeb 14, 2021

#1

+1

Baseball Both Basketball No sport

6 5 2 17 total = 30

13 out of 30

GuestFeb 14, 2021

#2

+193

0

Thank you! I ended up getting 22.

clairepowFeb 14, 2021

#1

+118703

+1

Hint:

It will have no solutions if the 2 lines are parallel. So the gradients must be the same.

MelodyFeb 14, 2021

#1

+506

+1

The angle of the n-sided polygon is $360-90-120-60=90$ degrees. Since a square is the only regular n-sided polygon with angles of 90 degrees, $\boxed{n=4}$