Your family is having pizza for dinner. Your mom asks you to
divide the 2 pizzas so everyone gets an equal amount. How
many servings of size can you make out of the 2 pizzas?

[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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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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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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Answer:

X= 4 or -4

Step-by-step explanation:

factor out from mathematical technique of difference of two squares.

Answer:

Step-by-step explanation:

26:45

Answer:

Step-by-step explanation:

1)Distribute. (-25a+180-20c)(25a+15b-20c). 2) Combine like terms. -625a+180+15b+400c.

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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y = -48 (1/x) is the variation equation.

A relationship between a set of values for one variable and a set of values for other variables is known as a variation.

One of the most fundamental concepts in mathematics is the idea of a function, in part because functions are a useful tool for describing and illuminating patterns.

given that

x and y vary inversely that means x = k 1/y or y = k 1/x.

so, x= k(1/y)

xy = k

from the question x=-12 when y=4

-12(4) = k

k = -48

now,

y = K (1/x)

y = -48 (1/x)

This is the variation equation.

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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

The coordinates of point b ( x, y) are ( - 5, 4 ).

The midpoint of the line ab is m( -1, -1 ).

The coordinates of the endpoints of the line are a( 3, -6) and b( x, y).

Now, the midpoint of the line with two endpoints A(x, y) and B( c, d) is given as:

m = [ ( x + c) / 2 , ( y + d) / 2 ]

Therefore,

m = [ ( 3 + x) / 2, ( - 6 + y ) / 2 ]

( -1 , - 1) = [ (3 + x / 2 ), ( -6 + y / 2 ) ]

Therefore,

(3 + x) / 2 = - 1

3 + x = - 2

x = - 2 - 3

x = - 5

And;

-6 + y / 2 = - 1

- 6 + y = - 2

y = - 2 + 6

y = 4

Therefore, ( x, y) = ( - 5, 4).

Therefore, the coordinates of point b are ( - 5, 4 ).

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Answer:

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

Step-by-step explanation:

An equation of the function in standard form is equal to 2x²/3 + 16x/3 + 8 = 0.

A quadratic function can be defined as a mathematical expression (equation) that can be used to define and represent the relationship that exists between two or more variable on a graph.

In Mathematics, the standard form of a quadratic equation is given by;

ax² + bx + c = 0

y = f(x) = ax² + bx + c

Next, we would formulate a system of equations from the data contained in the table above:

-2 = a(-3)² + b(-3) + c  ⇒ -2 = 9a - 3b + c      .......equation 1.

0 = a(-2)² + b(-2) + c  ⇒   0 = 4a - 2b + c      .......equation 2.

8 = a(0)² + b(0) + c  ⇒   8 = c                            .......equation 3.

Substituting the value of c into equation 1 and equation 2, we have:

-2 = 9a - 3b + 8    ⇒ 9a - 3b = -10    .......equation 4.

0 = 4a - 2b  + 8    ⇒ 4a - 2b = -8      .......equation 5.

Multiplying equation 5 by 1.5 and solving simultaneously, we have:

9a - 3b = -10

6a - 3b = -12

3a = 2

a = 2/3

For the value of b, we have:

4a - 2b = -8

4(2/3) - 2b = -8

8/3 - 2b = -8

2b = 8 + 8/3

2b = 32/3

b = 32/6

b = 16/3.

Substituting the values of a, b and c into the standard form of a quadratic equation:

ax² + bx + c = 0

2x²/3 + 16x/3 + 8 = 0

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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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Answer:

a) 1.999 moles He

b) 0.50 moles Cu

Step-by-step explanation:

Step-by-step explanation:

1 km = 1000 m = 0.621371 miles

so,

786.3 m = 0.7863 km (simply divide by 1000).

40 km/h = 40×0.621371 mph (as the reference of 1 hour stays the same).

= 24.8548 mph

always consider the direction of the conversion rate : are you going from a smaller to a larger unit, or from a larger to a smaller unit ?

if we would have said

1 m = 0.001 km

then we would have converted 786.3 m by multiplying it by 0.001 (also getting 0.7863 km as result, of course).

but because we defined it the other way around, we had to divide by 1000. it is clear that this is the same operation, yes ?

in the same way, if we would have said

1 mile = 1.60934 km

then we would have converted 40 km to miles by dividing 40 by 1.60934. again getting the same result, of course.

The factors of equation are -4 and -5.

Rewriting the equation -

x² + 9x + 20 = 0

ax² + bx + c = 0

The mentioned equation will be factorized using the given two rules -

1. sum of the factors will be equal to numeral in middle expression b (9x in this case)

2. product of factors will be equal to product of a and c (20 in this case)

Using above rules, the appropriate factors should be 5 and 4.

Representing the factors in mentioned equation -

x² + 4x + 5x + 20 = 0

Taking common factors

x(x + 4) +5(x + 4) = 0

(x + 4) (x + 5) = 0

x = -4, -5

Hence, the two factors of given equation are -4 and -5.

To check the answer, we will keep each value of x in  the equation -

When x = -4

x² + 9x + 20 = 0

(-4)² + 9(-4) + 20 = 0

16 -36 + 20 = 0

36 - 36 = 0

0 = 0

When x = -5

x² + 9x + 20 = 0

(-5)² + 9(-5) + 20 = 0

25 - 45 + 20 = 0

45 - 45 = 0

0 = 0

As both Left Hand Side and Right Hand Side of the equation are same for both values of x, the factors are correct.

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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

Answer:

4. Fixed and movable pulleys can be combined.

Step-by-step explanation:

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

Answer:

P = 3.75x - 7

Step-by-step explanation:

the perimeter (P) is the sum of the 3 sides of the triangle

P = 1.5x - 3 + 1.5x - 3 + 0.75x - 1 ← collect like terms

  = 3.75x - 7

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 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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Answer:

The lesser integer is 9.

Step-by-step explanation:

[tex]x + 4(x + 2) = 53[/tex]

[tex]x + 4x + 8 = 53[/tex]

[tex]5x + 8 = 53[/tex]

[tex]5x = 45[/tex]

[tex]x = 9[/tex]

If p and q vary inversely, and p=10 when q=-4 , the value of q will be 20 when p=-2.

A mathematical constant is a significant integer that has a steady value and is specified by a precise definition.

relationship between two variables in mathematics that may be represented by an equation where the product of their products equals a constant

Inverse variation is y = k. 1/x

given p = -4

q = 10

So, 10 = k. 1/-4

So, the value of k = -40

We can conclude that the value of constant variation is -40

Value of q, when p = -2

⇒ q = -40 / -2

⇒ q = 20

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Two expressions with equal sign is called as Equation, The value of - 5xV + 512 for x=-6 and J/ = 22 is 1172.

Two expressions with equal sign is called as Equation.

- 5xV + 512 for x=-6 and J/ = 22

Plug in values of x and V in above expression.

-5(-6)22+512

negative into negative is positive.

=660+512

=1172

There The value of expression - 5xV + 512 is 1172.

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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:

The scaling factor is 2: 1, the distance between Q and S is 3 units.

A scale factor is a number or conversion factor used to adjust the size of a figure without changing its shape. It is used to enlarge or reduce the size of an object. If the original dimensions are known, the scaling factor may be determined.

When describing a shape, you must define the amount to which it has been enlarged.

AC = 6

It signifies that the 6/3 = 2:1 scale factor has been preserved.

As a result, the total dimension of PQRS is half of ABCD.

So,

QS = 1/2 (BD)

Because BD = 6, (can be counted by graph grid)

So,

QS = 1/2 (5) = 3

Hence "Because the scale factor is 2: 1, the distance between Q and S is 3 units."

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