Expand each logarithm.

log 4 √47/5²

Answer:

X= 4 or -4

Step-by-step explanation:

factor out from mathematical technique of difference of two squares.

Answer:

-2x2 - 50

Step-by-step explanation:

First, you collect the like terms:

x2 - 3x2 + 5 x -8 -10

Do multiplication first:

x2 - 3x2 + (-40) - 10

Combine the terms:

-2x2 - 50

Answer:

5 star of you

Step-by-step explanation:

you are cool dictator

Answer:

a. -5 b. 23 c. 871 d. -5

Step-by-step explanation:

f(x) = x^2 + 6x and g(x) = x - 4

a). (fog)(3) = (x - 4)^2 + 6(x - 4)

= (3 - 4)^2 + 6(-1)

= 1 -6

= -5

b). (gof)(3) = ( x^2 + 6x) - 4

= (9 + 18) - 4

= 23

c). (fof)(3) = (x^2 + 6x)^2 + 6(x^2 + 6x)

= 27^2 + 27(6)

= 871

d). (gog)(3) = (x - 4) -4

= (3 - 4) - 4

= -5

The factoring of the expression x²-18 x+81 is (x-9)²

To solve this exercise, we have to follow the rules of factoring:

x²-18 x+81

1. Looking for the a and b number that state the following equations:

Analyzing that (a*b) is a positive number, it means that (a) and (b) have the same signs, and due to the results of (a+b) is negative the sings of (a) and (b) should be negative.  

The number could be:

a = - 9

b = - 9

Confirming:

a + b = -9 + (-9) = -18

a * b = (-9) * (-9) = 81

2. Rewriting the factored expression (x+a)(x+b) with the obtained values, we get:

(x-9)(x-9)

3. Rewriting as the square of a binomial, we get:

(x-9)²

Is a technique that consist of decomposition of a factor into a product of another factor, which when multiplied together give the original number.

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7x +2y= -17 is the result of adding these two equations 2x+3y=-5 and 5z-y = -12

What is equation ?

An equation is a mathematical statement with an 'equal to' symbol between two expressions that have equal values. For example, 3x + 5 = 15. There are different types of equations like linear, quadratic, cubic, etc.

given equations

2x+3y=-5

5z-y = -12

we have to add these two equations

Add 2x and 5x to get 7x

Add the y's

3y + (-1y)  OR 3y - y      

Which gives you 2y

add the (-5 and -12)

-12 + (-5) or -12 - 5 giving you -15

7x+2y=-17

Hence ,7x +2y= -17 is the result of adding these two equations 2x+3y=-5 and 5z-y = -12

the complete question is

What is the result of adding these two equations?

2x+3y= -5

5x - y = - 12

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Answer:

y = (1/6)x - 11

Step-by-step explanation:

y = (1/6)x + 7; Point: (12, -9)

                                 (x₁, y₁)

y - y₁ = m (x - x₁)

y - (-9) = (1/6) (x - 12)

y + 9 = (1/6)x - 2

    -9              -9

y = (1/6)x - 11

I hope this helps!

The height of an object dropped off a 350 foot cliff after 1.5 seconds is 314 feet

An equation is an expression that shows the relationship between two or more numbers and variables.

Let h(t) represent the height of an object after t seconds, hence:

h(t) = -16t² + 350

Hence:

At time t = 1.5:

h(1.5) = -16(1.5)² + 350 = 314 feet

The height of an object dropped off a 350 foot cliff after 1.5 seconds is 314 feet

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The required factors of the expression, 4x² - 9 are 4, x, x, -3, 3.

What are the factors of an expression?

Any number or expression that completely divides another number or expression is called as the factor of an expression. These can be positive or negative depending upon the given expression.

Calculating the factor of expression. 4x² - 9

Given an expression, 4x² - 9

Terms of an expression are 4x², - 9

Further breaking the terms to factors, we obtain,

4, x, x, -3, 3

Hence, the required factors of the expression, 4x² - 9 are 4, x, x, -3, 3.

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Answer:

2850 kilograms

Step-by-step explanation:

step 1)

change the percent to a decimal

by moving the decimal over to spots to the left

95.0% to 0.95

step 2

Multiply tge decimal with the amount you were given

0.95 × 3000 = 2850

If you want to double check, do this

2850 ÷ 3000 = 0.95

Answer: B

Step-by-step explanation: If you look at the phrase m<1 + m<2=180 - m<3 you realize that you can subtract m<3 from both sides so the equation becomes m<1 + m<2 + m<3=180

The interval in which the function has positive average rate of change is x > √3

The function is given as

f(x) = x^3 - 9x

Differentiate the above function to determine the average rate of change

So, we have

f'(x) = 3x^2 - 9

Set the function to greater than 0

So, we have

3x^2 - 9 > 0

Add 9 to both sides

3x^2 > 9

Divide by 3

x^2 > 3

Take the square root of both sides

x > √3

Hence, the interval in which the function has positive average rate of change is x > √3

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Answer:

4. Fixed and movable pulleys can be combined.

Step-by-step explanation:

The equation of the parabola that has the same shape as  [tex]f(x) = -8x^2[/tex], but with the vertex (4,9) is  [tex]f(x) = -8(x - 4)^2 + 9.[/tex]

The general equation of a parabola is given by

y = a(x – h)² + k

or x = a(y – k)² + h.

Here (h, k) denotes the vertex.

Now it is given that,

The parent function is [tex]f(x) = -8x^2[/tex]

The vertex of parabola is given as (4,9)

Thus, we have

h = 4

k = 9

Now since, the transformations form of a parabola is given as:

[tex]f(x) = a(b(x - h)) + k[/tex]

where:

a = vertical distance

b = horizontal distance

h = horizontal shift, x-coordinate of vertex

k = vertical shift, y-coordinate of vertex

Therefore, transformed equation of parabola is given as,

[tex]f(x) = -8(x - 4)^2 + 9[/tex]

this is the required equation of parabola with the vertex (4,9).

Thus, the required equation of the parabola that has vertex (4,9) is [tex]f(x) = -8(x - 4)^2 + 9[/tex].

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If two positive numbers are multiplied or divide then the result is always positive but if we take one positive and one negative number and then after multiply or divide we get always negative number.

What is Number system?

A system of writing to express the number with positive or negative sign is called number system.

Now, There are many applications for the use of positive and negative numbers to represent mathematical ideas.

⇒ Negative and positive numbers are used to describe the distance from a reference point.

⇒ If two positive numbers are multiplied or divide then the result is always positive but if we take one positive and one negative number and then after multiply or divide we get always negative number.

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The inverse of [tex]f(x) = \sqrt{x - 4}[/tex] is [tex]f^{-1}(x) = x^{2} +4[/tex] that is option G) is correct

According to the question we have been given the function which is :

                  [tex]f(x) = \sqrt{x - 4}[/tex]

To find the inverse of this function we will solve this step by step

First we will convert it into equation form

    [tex]y = \sqrt{x - 4}[/tex]

Then we will interchange the variables x and y

 [tex]x = \sqrt{y - 4}[/tex]

Then we will solve for y, we get

Taking square on both sides of the equation, we get

[tex]x^{2} = y - 4[/tex]

[tex]y = x^{2} + 4[/tex]

Now replacing variable y with [tex]f^{-1} (x)[/tex] , we get

[tex]f^{-1}(x)[/tex] = [tex]x^{2} +4[/tex]

Thus the inverse of [tex]f(x) = \sqrt{x -4}[/tex] is [tex]f^{-1} = x^{2} +4[/tex] that is option G) is correct

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The number of cycles of the periodic function is 3.75 cycles if the period of a periodic function is 8 s option (G) 3.75 is correct.

It is defined as a special type of relationship, and they have a predefined domain and range according to the function every value in the domain is related to exactly one value in the range.

It is given that:

The period of a periodic function is 8 s

From the question:

8n = 30

n = 30/8

n = 3.75 cycles

Thus, the number of cycles of the periodic function is 3.75 cycles if the  period of a periodic function is 8 s option (G) 3.75 is correct.

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Answer: ob false

Step-by-step explanation: look at a calculator and it will show 8 is not a 5 is not 8 times five is 40 hope this helps

[tex]2^{3x-4}=5[/tex] => x = 2.106 , by properties of logarithm.

Given: [tex]2^{3x-4}=5[/tex]

Taking logarithm on both sides:

log ([tex]2^{3x-4}[/tex]) = log (5)

=> (3x - 4) log (2) = log (5) (As [tex]log_b(a)^m = m (log_b(a))[/tex])

=> 3x - 4 = [tex]\frac{log5}{log2}[/tex]

=> 3x - 4 = [tex]\frac{0.698}{0.301}[/tex]

=> 3x - 4 = 2.318

=> 3x = 6.318

=> x = 2.106

Hence, [tex]2^{3x-4}=5[/tex] => x = 2.106, by properties of logarithm.

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Answer:

[tex](x,y,z)=\left(-\dfrac{1}{11},\dfrac{21}{11},\dfrac{21}{11}\right)[/tex]

Step-by-step explanation:

Given:

[tex]-2x+3y-z=-23x+y=4-2y+2z=4[/tex]

Therefore:

[tex]\begin{cases}-2x+3y-z=4\\-23x+y=4\\4-2y+2z=4\end{cases}[/tex]

Rewrite Equation 2 to make x the subject:

[tex]\implies -23x+y=4[/tex]

[tex]\implies -23x+y-y=4-y[/tex]

[tex]\implies -23x=4-y[/tex]

[tex]\implies \dfrac{-23x}{-23}=\dfrac{4}{-23}-\dfrac{y}{-23}[/tex]

[tex]\implies x=-\dfrac{4}{23}+\dfrac{y}{23}[/tex]

Rewrite Equation 3 to make z the subject:

[tex]\implies 4-2y+2z=4[/tex]

[tex]\implies 4-2y+2z-4=4-4[/tex]

[tex]\implies -2y+2z=0[/tex]

[tex]\implies -2y+2z+2y=0+2y[/tex]

[tex]\implies 2z=2y[/tex]

[tex]\implies \dfrac{2z}{2}=\dfrac{2y}{2}[/tex]

[tex]\implies z=y[/tex]

Substitute the found expressions for y and z into Equation 1 and solve for y:

[tex]\implies -2x+3y-z=4[/tex]

[tex]\implies -2\left(-\dfrac{4}{23}+\dfrac{y}{23}\right)+3y-y=4[/tex]

[tex]\implies \dfrac{8}{23}-\dfrac{2}{23}y+2y=4[/tex]

[tex]\implies \dfrac{8}{23}+\dfrac{44}{23}y=4[/tex]

[tex]\implies \dfrac{8}{23}+\dfrac{44}{23}y- \dfrac{8}{23}=4- \dfrac{8}{23}[/tex]

[tex]\implies \dfrac{44}{23}y=\dfrac{84}{23}[/tex]

[tex]\implies \dfrac{44}{23}y \cdot \dfrac{23}{44}=\dfrac{84}{23}\cdot \dfrac{23}{44}[/tex]

[tex]\implies y=\dfrac{84}{44}[/tex]

[tex]\implies y=\dfrac{21}{11}[/tex]

Therefore, as y = z:

[tex]\implies z=\dfrac{21}{11}[/tex]

Substitute the found value of y into the found expression for x and solve for x:

[tex]\implies x=-\dfrac{4}{23}+\dfrac{y}{23}[/tex]

[tex]\implies x=-\dfrac{4}{23}+\dfrac{\frac{21}{11}}{23}[/tex]

[tex]\implies x=-\dfrac{4}{23}+\dfrac{21}{253}[/tex]

[tex]\implies x=-\dfrac{1}{11}[/tex]

Therefore, the final solution is:

[tex](x,y,z)=\left(-\dfrac{1}{11},\dfrac{21}{11},\dfrac{21}{11}\right)[/tex]

The equivalent expression of the expression given as 2π(3.5)(6)+ 2π3.52 in terms of π is 49.04π

The expression is given as

2π(3.5)(6)+ 2π3.52

Rewrite the above expression properly as follows

So, we have

2 * π * (3.5) * (6) + 2 * π * 3.52

Evaluate the products of the factors of the above expression

So, we have

42 * π + 7.04 * π

Evaluate the products of the constants in the above expression

So, we have

42π + 7.04π

Evaluate the sum in the above expression

So, we have

49.04π

Hence, the equivalent expression of the expression given as 2π(3.5)(6)+ 2π3.52 in terms of π is 49.04π

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Answer:

It will take 20 minutes  for both buckets to have the same amount of water

Step-by-step explanation:

In design 8, number of hearts = 2 × 8 × 8 = 128 and number of diamonds is 32.

a) The offered designs show that the height always remains 1 cube, but the length and width both grow by 1 cube.

In design 1,

Number of hearts = 2 and number of diamonds = 4

In design 2,

Number of hearts = 8 and number of diamonds = 8

In design 3,

Number of hearts = 18 and, number of diamonds = 12

We have hearts at top and bottom and diamonds along the side walls.

b)

In deisgn 1  l× b × h = 1 × 1 × 1

Hearts = 2 = 2 × l ×b

Diamonds = 4 = l × 4

In deisgn 2 1 × b × h = 2 × 2 × 1

Hearts = 8 = 2 × l × b

Diamonds = 8 = l × 4

In deisgn 3 l × b × h = 3 × 3 × 1

Hearts = 18 = 2 × l × b

Diamonds = 12 = l × 4

So for any design,

number of hearts = 2 × l × b  

number of diamonds = l × 4

c)

Company have 4 times hearts as diamonds.

So we can write,

2 × l × b = 4(l × 4) =

We can see in every case l=b. so

2 × l^2 = 16l

l = 8

So company have to advertise 8 × 8 × 1  design and put it on sale.

In design 8,

number of hearts = 2 × 8 × 8 = 128 and number of diamonds = 4 × 8 = 32.

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It should be noted that the slope shows that the rate of change in elevation rise at first but later reduced.

The total change in elevation is 10500 feet

It should be noted that from the table, the time and the elevation has been given.

Therefore, the total elevation will be the addition of all the values given. This will be 10500 feet.

Also, the slope shows that the rate of change in elevation rise at first but later reduced.

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Answer:

the numbers are 27,28

Step-by-step explanation:

Let the numbers be x,x+1

The sum is 55

⟹x+x+1=55

2x+1=55

2x+54

x=27

So the numbers are 27,28

Answer:314

Step-by-step explanation:

radius=diameter divided by 2

area=[tex]\pi radius^{2}[/tex]

a=[tex]\pi 10^{2}[/tex]

a=314.16

rounded to the nearest whole

a=314

The volume of water in the vase is 103.5π inches³.

Diameter (d) of Cylindrical vase = 6 inches

Radius (r) of Cylindrical vase = d/2 = 3 inches

Height (h) of Cylindrical vase = 12 inches

Volume of Cylindrical vase (V) = πr²h

                                                     = π × 3 × 3 × 12

                                                     = 108π inches³

As we know that at the bottom of the vase, there are 7 marbles,

and,  Diameter of Marbles = 3 inches

         Radius of Marbles = 1.5 inches

Marble is a shape of sphere, so

The Volume of Marble = 4/3 (πr³)

                                       = 4/3 ( π × 1.5 × 1.5 × 1.5)

                                       = 4/3 ( 3.375π)

                                       = 4.5π inces³

To find the volume of water in the vase, we have to subtract the volume marbles from the volume of cylindrical vase,

The volume of water in the vase = 108π inches³ - 4.5π inces³

                                                       = 103.5π inches³

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Answer:

n(3in)2(12in) - 7(4/3n(1.5in)3)

Step-by-step explanation: