A vending machine needs to be programmed to accept any combination of coins totaling 0.75, excluding pennies, and half dollars. How many combinations are possible?

 1.1      Factoring:  x3-81 

Theory : A difference of two perfect cubes,  a3 - b3 can be factored into
              (a-b) • (a2 +ab +b2)

Proof :  (a-b)•(a2+ab+b2) =
            a3+a2b+ab2-ba2-b2a-b3 =
            a3+(a2b-ba2)+(ab2-b2a)-b3 =
            a3+0+0+b3 =
            a3+b3

Check :  81  is not a cube !! 
Ruling : Binomial can not be factored as the difference of two perfect cubes

 1.2    Find roots (zeroes) of :       F(x) = x3-81
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q  then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  1  and the Trailing Constant is  -81. 

 The factor(s) are: 

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,3 ,9 ,27 ,81 

 Let us test ....

Polynomial Roots Calculator found no rational roots

 2.1      Solve  :    x3-81 = 0 

 Add  81  to both sides of the equation : 
                      x3 = 81 
When two things are equal, their cube roots are equal. Taking the cube root of the two sides of the equation we get:  
                      x  =  ∛ 81  

 Can  ∛ 81 be simplified ?

Yes!   The prime factorization of  81   is
   3•3•3•3  
To be able to remove something from under the radical, there have to be  3  instances of it (because we are taking a cube i.e. cube root).

∛ 81   =  ∛ 3•3•3•3   =
                3 • ∛ 3 

The equation has one real solution
This solution is  x = 3 • ∛3 = 4.3267 

Answer:

The initial value is  

✔ 1/3

The simplified base is  

✔ 27

The domain is  

✔ all real numbers

The range is  

✔ y > 0

Step-by-step explanation:

The approximate solution for the given system of equations y = 2x2 – 2 and y = –0.5x + 4 is C. x = 1. This is the value of x where the two graphs intersect, thus making both equations true.

The question is asking for the solution to the system of equations given by y = 2x2 – 2 and y = –0.5x + 4. Solving a system of equations means finding the value of x that makes both equations true. In other words, we're looking for the point where the two graphs intersect.

To find this value, we can set the two equations equal to each other and solve for x:

2x2 – 2 = -0.5x + 4

This simplifies to 2x2 + 0.5x - 6 = 0. Solving this quadratic equation can give us multiple solutions for x, but since the question is multiple choice, we can check which of the given options is a solution. Plugging the options into the equation reveals that option C. x = 1 is the correct answer, satisfying the equation.

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In his piggy bank, Dan has 10 nickels, 20 dimes and 5 quarters.

Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.

Assume that he had [x] nickels, [y] dimes and [z] quarters. Then we can write -

x = y + 5

x + y + z = 35

x/20 + y/10 + z/4 = 6

Now, the equation → x/20 + y/10 + z/4 = 6, can be written as → (x + 2y + 5z) = 120

We can write, the equations as -

x = y + 5

x + y + z = 35

x + 2y + 5z = 120

Solving the simultaneous equations, we get -

y + 5 + y + z = 35

2y + z = 30         Eq [1]

y + 5 + 2y + 5z = 120

3y + 5z = 115        Eq [2]

Plot Eq[1] and Eq[2] on graph. The point of intersection of both will give the values of y and z.

z = 20

y = 5

We can then find [x] as -

x = y + 5 = 10

x = 10

Therefore, in his piggy bank, Dan has 10 nickels, 20 dimes and 5 quarters.

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8*3 = 24 + 28 = 52 total square feet

he is painting at 8 sq m per hour

so A(t) = -8t +52

Answer:

[tex]A(t)=-8t+52[/tex]

Step-by-step explanation:

Let

A(t)------> the area to paint

t------> the time in hours

Step 1

Find the total area of the room

[tex]A=8*3+28=52\ m^{2}[/tex]

Step 2

Find the linear equation that represent the situation

[tex]A(t)=-8t+52[/tex]

The sentence 'k is less than zero' can be translated into the inequality 'k < 0', where 'k' can be any value that is less than zero. The inequality for the word sentence 'k is less than zero' is k < 0.

The sentence 'k is less than zero' can be translated into an inequality as 'k < 0'. This is an inequality to represent a situation where 'k', a variable, takes values that are less than zero. It's a way of understanding the relationship of 'k' to zero in mathematical terms. For example, -1, -2, -3, etc. are all numbers that meet this less than zero inequality because they are less than zero.

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To determine two numbers where their difference is 16, and their ratio is 5:7, we let the smaller number be 5x and the larger one be 7x, solve for x with the equation 7x - 5x = 16, and find that the numbers are 40 and 56.

To find two numbers with a difference of 16 and a ratio of 5:7, first let's denote the smaller number as 5x and the larger number as 7x. The difference between the two numbers is 16, so we can write the equation 7x - 5x = 16.

Solving for x gives us 2x = 16 or x = 8. Thus, the two numbers we are looking for are 5x = 5 × 8 = 40 and 7x = 7 × 8 = 56.

Check: The difference between 56 and 40 is indeed 16, and the ratio of 40 to 56 simplifies to 5:7, confirming our solution.

The volume and the surface area of the rectangular prism will be 66mm and 166mm² respectively.

Length = 11m

Width = 1mm

Height = 6mm

The volume of the rectangular prism will be:

= Length × Width × Height

= 11mm × 1mm × 6mm

= 66mm

The surface area of the rectangular prism will be:

= 2(wl + hl + hw)

= 2(11 + 66 + 6)

= 2(83)

= 166

Therefore, the surface area of the rectangular prism will be 166mm²

In conclusion, the volume and the surface area of the rectangular prism will be 66mm and 166mm² respectively.

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Yeah it's B:17.8 I agree with you

Answer:

Inverse statement : "If you don't lift weights, then you will not be strong".

Step-by-step explanation:

If a statement is defined as

p → q : If p is true, then q is true.

then the inverse of the given statement is

¬ p → ¬ q : If p is not true, then q is not true.

The given statement is "If you lift weights then you will be strong".

We need to find the inverse of the given statement.

Here,

p = you lift weights

q = you will be strong

The inverse of the given statement is "If you not lift weights, then you will not be strong".

volume = l x w x h

1.5 x 1.5 x 1.5 = 3.375 cubic feet

3.375 * 57 = 192.375 pounds

By using what we know about intersecting lines, we will see that the correct options are:

In an intersection of two lines, if two angles are adjacent, then their measures must add to 180°. And if the angles are not adjacent, then the angles have the same measure.

a) 1 and 4 are not adjacent, so these angles have the same measure.

so if m<1 = 43, then m<4 = 43

The correct option is B.

b) 1 and 5 are equivalent (where we assume that JL and MP are parallel, so all the angles formed in the two intersections are equal).

Then if m<1 = 50,  m<5 = 50.

So the correct option is A.

c) The corresponding angles occupy equivalent positions.

For example:

etc.

The correct option is B.

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Answer:

Teachers 4 and Students 100, Teachers 5 and Students 125

Step-by-step explanation:

Going by the chart, by each teacher, the students are increasing by 25. The only ones to follow this pattern is 4:100 and 5:25