Suppose we have tossed two fair dice. given that the sum is 8, what is the probability that one of the dice is 6?

10. Determine the open interval(s) on which the function f(x) = 12x - x^3 is increasing.

Gre4nikov [31]

Answer:

$\large\boxed{x\in(-2,\ 2)}$

Step-by-step explanation:

$f(x)=12x-x^3$

Calculate the derivative of the function:

$f'(x)=\left(12x-x^3\right)'=(12x)'-\left(x^3\right)'=12-3x^2$

Calculate the zeros of the derivative:

$f'(x)=0\Rightarrow12-3x^2=0\qquad\text{add}\ 3x^2\ \text{to both sides}\\\\3x^2=12\qquad\text{divide both sides by 3}\\\\x^2=4\to x=\pm\sqrt4\\\\x=\pm2$

Let's check the derivative sign:

$f'(x)>0\iff12-3x^2>0\qquad\text{use the distributive property}\\\\-3(x^2-4)>0\\\\-3(x^2-2^2)>0\qquad\text{use}\ a^2-b^2=(a-b)(a+b)\\\\-3(x-2)(x+2)>0\\^{\qquad x=2\qquad x=-2$

Sketch a parabola open down

(before the brackets there is the number -3 < 0)

Look at the picture.

Where the derivative is negative, the function decreases.

Where the derivative is positive, the function increases.

Therefore your answer is:

$x\in(-2,\ 2)$

3 0

4 years ago

A costume designer has 36 bat buttons and 18 black cat buttons she must use to create costumes. In her costumes, she used both b

Advocard [28]

A costume designer has 36 bat buttons and 18 black cat buttons she must use to create costumes

To find the largest number of costumes we take GCF

GCF gives the largest and greatest number of grouping

WE find GCF of 36 and 18

1, 2, 3, 6, 9, 18 are the factors of 18

1, 2, 3, 4, 6, 9, 12, 18, 36 are the factors of 36

Greatest common factor GCF is 18

So , 18 is the largest number of costumes she can make.

7 0

4 years ago

Simplify by rationalizing the denominator.<br> 15√2 / √270

ladessa [460]

$\frac{ \sqrt{15} }{3}$ ✅

Step-by-step explanation:

$\frac{15 \sqrt{2} }{ \sqrt{270} } \\ \\ = \frac{15 \sqrt{2} }{ \sqrt{270} } \times \frac{ \sqrt{270} }{ \sqrt{270} } \\ \\= \frac{15 \sqrt{2} \times \sqrt{270} }{270} \\\\ (∵ \sqrt{a} \times \sqrt{a} = a) \\\\ = \frac{15 \sqrt{540} }{270} \\ \\= \frac{ \sqrt{2 \times 2 \times 3 \times 3 \times 3 \times 5} }{18} \\ \\ = \frac{ \sqrt{ {2}^{2} \times {3}^{2} \times 15 } }{18} \\ = \frac{2 \times 3 \sqrt{15} }{18} \\ \\ = \frac{6 \sqrt{15} }{18} \\ \\ = \frac{ \sqrt{15} }{3}$

$\large\mathfrak{{\pmb{\underline{\orange{Mystique35 }}{\orange{♡}}}}}$

7 0

3 years ago

Read 2 more answers

I'd really appreciate it if anyone could help! :)

Ahat [919]

Answer:

The given table represents a function f(x) = y = 2x+3

The upper graph

Step-by-step explanation:

We will write the equation of the right line in the explicit (slope) form

y = ax+b

We will take two points from the given table

A(-1,1) and B(0,3) and insert their coordinates in the equation y=ax+b and get

A(-1,1) => 1 = (-1)*a+b => -a+b=1

B(0,3) => 3=0*a+b => 3= 0+b => b=3

Now we will insert b=3 in the equation -a+b=1 and get

-a+3=1 => a=3-1 => a=2

Now we will insert a=2 and b=3 in the equation y=ax+b and get

equation of the function

y = 2x+3 = f(x) The upper graph.

Good luck!!!

5 0

3 years ago

Triangle ABC has verrocktes A (2,-1) B ( -3,0) and C ( -1,4) . Find the vertices of the image of triangle and after a translatio

belka [17]

Answer:

A(2,1) , B(-3,2) , C(-1,6)

Step-by-step explanation:

A translation of 2 units up means the y values will be shifted up 2 units

A (2,-1) -> A(2,1)

B (-3,0) -> B(-3,2)

C (-1,4) -> C(-1,6)

3 0

3 years ago