Which point is a solution to the system?

Answer:  

Dimensions = 20 inches by 20 inches by 80 inches

The floor is 20 inches by 20 inches. The height is 80 inches.

The minimized cost is $2400.

========================================================

Work Shown:

h = height of the box, in inches

x = side length of the square base, in inches

x^2 = area of the floor = area of the ceiling

x^2h = volume of the rectangular box

x^2h = 32000

h = 32000/(x^2)

C = total cost in dollars

C = 0.25*(area of the sides) + 1.00*(area of the floor and ceiling)

C = 0.25*(xh+xh+xh+xh) + 1.00*(x^2+x^2)

C = 0.25*4xh + 1.00*2x^2

C = xh + 2x^2

C = x*(32000/(x^2)) + 2x^2

C = (32000/x) + 2x^2

C = (32000/x) + (x*2x^2)/x

C = (32000/x) + (2x^3)/x

C = (32000+2x^3)/x

The goal is to make C the smallest possible, aka we want to minimize it.

Visually we want the lowest point on the cost curve.

We have two options to get this task done:

I'll assume your teacher hasn't gone over calculus at this point. I'll go for option 1 mentioned above.

Use a graphing tool like GeoGebra to plot out the cost function curve. See the diagram below. The lowest point on this curve is at (20,2400).  I used the "min" function to determine this lowest point. Keep in mind that x > 0.

This lowest point indicates to us that x = 20 causes C(x) to be the smallest at $2400. This is the minimized cost.

Therefore, the square base should be 20 inches by 20 inches. The height should be:

h = 32000/(x^2) = 32000/(20^2) = 80 inches

The box should be 20 inches by 20 inches by 80 inches.

------------------------

Check:

Volume = length*width*height = 20*20*80 = 32000 cubic inches

This helps verify the answer.

Answer:

Width = 20 inches

Length = 20 inches

Height = 80 inches

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{6.7 cm}\underline{Volume of a square-based rectangular box}\\\\$V=x^2h$\\\\where:\\\phantom{ww} $\bullet$ $x$ is the side length of the base.\\\phantom{ww} $\bullet$ $h$ is the height.\\\end{minipage}}[/tex]

Given the volume of the square-based rectangular box is 32,000 in³, substitute this into the equation and rearrange to isolate h:

[tex]\implies x^2h=32000[/tex]

[tex]\implies h=\dfrac{32000}{x^2}[/tex]

[tex]\boxed{\begin{minipage}{7.4 cm}\underline{Surface Area of a square-based rectangular box}\\\\$S=2x^2+4xh$\\\\where:\\\phantom{ww} $\bullet$ $x$ is the side length of the base.\\\phantom{ww} $\bullet$ $h$ is the height.\\\end{minipage}}[/tex]

Given:

Create an equation for the total cost, C, of the materials based on the equation for the surface area:

[tex]\implies C=(1)2x^2+(0.25)4xh[/tex]

[tex]\implies C=2x^2+xh[/tex]

Substitute the expression for h into the equation for cost to create an equation for C in terms of x:

[tex]\implies C=2x^2+x\left(\dfrac{32000}{x^2}\right)[/tex]

[tex]\implies C=2x^2+\dfrac{32000}{x}[/tex]

[tex]\implies C=2x^2+32000x^{-1}[/tex]

To find the value of x that would minimize the cost, differentiate the equation for cost:

[tex]\implies \dfrac{\text{d}C}{\text{d}x}}=4x-32000x^{-2}[/tex]

[tex]\implies \dfrac{\text{d}C}{\text{d}x}}=4x-\dfrac{32000}{x^{2}}[/tex]

[tex]\implies \dfrac{\text{d}C}{\text{d}x}}=\dfrac{4x^3-32000}{x^{2}}[/tex]

Set the differentiated equation to zero and solve for x:

[tex]\implies \dfrac{4x^3-32000}{x^{2}}=0[/tex]

[tex]\implies 4x^3-32000=0[/tex]

[tex]\implies 4x^3=32000[/tex]

[tex]\implies x^3=8000[/tex]

[tex]\implies x=20[/tex]

Therefore, the side lengths of the base of the box that would minimize the cost are 20 inches.

To find the height of the box that would minimize the cost, substitute the found value of x into the expression for height:

[tex]\implies h=\dfrac{32000}{20^2}[/tex]

[tex]\implies h=\dfrac{32000}{400}[/tex]

[tex]\implies h=80\; \rm in\;(2\:d.p.)[/tex]

Therefore, the dimensions of the box that would minimize cost are:

B1 should be smaller in order to spread out the explanatory components in order to test a B1 hypothesis or generate confidence intervals around a B1 hypothesis.

Though an explanatory variable is a hypothesis that is predicted to be able to explain the research outcomes, a response variable demonstrates the impact that is anticipated to come from the explanatory variable. The regression line has a slope that is B1.

In other circumstances, the alternative hypothesis is compared against the null hypothesis to determine whether the assertion that the population slope is not equal to Zero is valid. The value of b1 is to be construed as the average result altering when the explanatory variable is increased by one unit.

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C = 12k, where C represents the cost on the electrical bill and k represents the electricity used is a relationship that varies directly. Option C is correct.

proportionality is defined as between two or more sets of values, and how these values are related to each other in the sense are they directly proportional or inversely proportional to each other.

here,
A standard directly proportional relationship is given as,
y = kx

Now,
comparing the above equation with the equation among the option,
If the equation in options matches with the above equation, so that equation will be a relationship that varies directly.

Thus, C = 12k, where C represents the cost on the electrical bill and k represents the electricity used is a relationship that varies directly. Option C is correct.

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The likelihood that the first question will be true is Equally likely and unlikely. That is option C

Probability is defined as the possibility of an outcome of an event which may likely or unlikely occur.

The true and false quiz given by the teacher;

P(true) = 0.5 for each question.

Therefore, False = 0.5 for each question making the probability of the first question being true to be equally likely and unlikely.

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Answer:

20 Stickers

Step-by-step explanation:

Set up for the problem

22+16= 38

38-18=20

So, Overall=20...

Answer:

8) AD = 23

9) m<B = 63

10) G(1,-5)

-------------------------------

8)

AD = BC

Thus, x + 21 = 12x - 1

11x = 22

x = 2

So, AD = 23

9)

consecutive angles are supplementary, sum to 180

y/2 + y - 9 = 180

3/2 y  = 189

y = 126

so, m<B = 63

10)

Diagonals bisect each other. The midpoint of DF = (-1.5, 0.5)

-1.5 = 2+x/2

x = 1

0.5 = 6+y/2
y = -5

G(1,-5)

2x-1 with a translation 1 unit left followed by a reflection in the x-axis is -2x -1.

What is translation and reflection?

Flipping an object across a line without causing it to change in size or shape is called reflection. An object can rotate around a fixed point without changing its size or shape. Changing a figure's size, shape, or orientation does not constitute translation.

Reflections and translations are two of the most popular types of transformations. An object moves from one location to another through translation, remaining the same size and orientation.

One unit left means  -1 shift on x-axis leads to F(x+1)

Reflection on x-axis means rotating the graph upside down on x-axis which leads -F(x+1)

-F(x + 1) = -2(x + 1) + 1 = -2x - 2 + 1 = -2x -1

Hence, 2x-1 with a translation 1 unit left followed by a reflection in the

x-axis is -2x -1.

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The distance between point G and H is 4√2 units51.96 unit²

An equation is a expression that shows the relationship between numbers and variables.

The distance between two points A(x₁, y₁) and B(x₂, y₂) on the origin is:

[tex]AB=\sqrt{(y_2-y_1)^2+(x_2-x_1)^2}[/tex]

Let us take point G as out reference point (origin)

Hence:

G = (0, 0) and H = (4, -4)

[tex]GH=\sqrt{(-4-0)^2+(4-0)^2}=4\sqrt{2}\ units[/tex]

The distance is 4√2 units

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Oki and Stephen can only produce 0.75 bags of trail mix in total.

Two numbers can have a division symbol (") added between them to indicate that they have been divided. Therefore, we can write 36 6 if we need to demonstrate the division of 36 by 6. A fractional representation is 366, which we can also use.

Compared to multiplication, division is the opposite. When you divide 12 into three equal groups, you get four in each group if three groups of four add up to 12, which they do when you multiply.

The primary objective of division is to count the number of equal groups that are created or the number of individuals in each group after a fair distribution.

According to question:-

3/3 : 4/2 =1:2

90/120 = 0.75.

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Answer:

The sequence is geometric.

Step-by-step explanation:

Arithmetic Sequence:

A arithmetic sequence is a sequence of numbers in which the next term is calculated by adding some constant amount to the current term. It can be seen almost as a linear equation, but "x" is always a whole number, that starts at zero and increases by one.

We can check if a sequence is arithmetic or not, by subtracting any term from the next term. This amount should be constant for all terms if a sequence is arithmetic. This constant amount can be seen as the slope, how much it's changing by each term.

Geometric Sequence:

A geometric sequence is a sequence of numbers in which the next term is calculated by multiply by some constant amount and the current term. It can be seen almost as a exponential equation, but "x" is always a whole number, that starts at zero and increases by one.

Using the definition above, it can also be thought of as the current term being equal to the previous term multiplied by a constant amount. This means if we divide any term in the sequence by the previous term, this should be a constant amount, no matter which term we use.

Now that we know what the definitions of a geometric and arithmetic sequence as well as how to check if a sequence is either, we can now apply this knowledge to the problem. Let's start by checking if the sequence is arithmetic.

We can start by using the first term "45" and subtracting it from the second term "15", which gives us "-30". This means we had to "add" "-30" to get the second term. If this sequence is arithmetic, this means we could add this to the second term and get the third term. If we add "-30" to "15" we get "-15" which is not equal to the next term. So the amount that it's changing by is not constant, meaning this sequence is not arithmetic.

Now let's check if the sequence is geometric. Each term can be defined as the previous term multiplied by some constant amount. So if we divide any term by it's previous term we get this amount that it had to be multiplied by which should be constant. We cannot start with the first term, since there is no previous term before the first term. So let's start with the second term: "15", now let's divide it by the previous term, which is the first term: "45" [tex]\frac{15}{45} = \frac{1}{3}[/tex]. If this sequence is geometric, we can multiply the second term by this 1/3 and get the next term, which is the third term. If we multiply "15" (the second term) by "1/3" we get [tex]\frac{15}{3}[/tex] which is equal to "5", which is equal to the third term. So this sequence appears to be geometric.

The solution is

a) The motor boat traveled a displacement of

b) The velocity of the tugboat

c) The negative distances and velocities means the boat is traveling down a river

Velocity is a vector expression of the displacement that an object or particle undergoes with respect to time. It is a vector quantity. Velocity is the change in displacement of an object with respect to time

Velocity = Displacement / Time

Given data ,

Some boats are traveling up and down a river

So , when the boat is traveling up a river , the velocity will be positive and when the boat is traveling down a river , the velocity will be negative

a)

The velocity of motor boat = - 3.4 mph

The time taken by the boat = 0.75 hours

Velocity = Displacement / Time

Substituting the values in the equation , we get

Displacement = Velocity x Time

Displacement D = - ( 3.4 ) x 0.75

On simplifying the equation , we get

Displacement D = -2.55 miles

Therefore , the boat traveled -2.55 miles down

b)

The displacement of the tugboat D = -1.5 miles

The time taken by the tugboat T = 0.3 hours

So , velocity of the tugboat V = ( -1.5 ) / 0.3

On simplifying the equation , we get

Velocity V = -5 mph

Therefore , the velocity of the tugboat is -5 miles per hour

c)

When the boat is traveling down a river , the velocity will be negative

Hence , the equations are solved

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When the pyramid is cut out of the prism, it has a volume of 36 cubic meters and the same base and height.

It is also known as the object's capacity. The fundamental equation for volume is length, width, and height, as opposed to the fundamental equation for the area of a rectangular shape, which is length, breadth, and height. The math is the same regardless of how you refer to the different dimensions. For example, you can substitute "depth" for "height." Volume is used to describe an object's capacity.  Volume can also be used to describe how much space a three-dimensional object occupies.

given

The ratio of their volumes is always one to three

Since the volume of a prism is 108 cubic meters,

a pyramid's volume is 1/3 * 108, or 36 cubic meters.

When the pyramid is cut out of the prism, it has a volume of 36 cubic meters and the same base and height.

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The graph of the new trapezoid is attached

Transformation is the movement of a point in the coordinate plane. Types of transformation are rotation, dilation, reflection and translation.

Dilation is the increase or decrease in size of a figure by a scale factor. The rule for dilation is:

(x, y) ⇒ (kx, ky)

The vertices of the trapezoid is P(1, -2), Q(2, -2), R(2, 1) and S(-2, 1)

If the trapezoid is dilated by a scale factor of 4 about the origin, the new points are:

P'(4, -8), Q'(8, -8), R'(8, 4) and S'(-8, 4)

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The approximation of the area of f(x) = x² + 2x, from x = 2 to x = 10, is given as follows:

320 units squared.

The area under a curve is approximated using the definite integral of the function that defines the curve within it's bounds.

As it is specified that the area should be approximated using rectangles, a Left Riemann Sum will be used to approximate the area.

First we must obtain the Delta value, which is the difference between the bounds of the integral, divided by the number of intervals, thus, considering four intervals, we have that:

[tex]\Delta_x = \frac{10 - 2}{4} = 2[/tex]

Then the values of x at which the numeric values are calculated are obtained as follows:

[tex]x_i = a + \Delta_x(i - 1)[/tex]

Thus the values of x are of:

The numeric values are given as follows:

Then the definite integral is obtained as follows:

[tex]\sum_{i = 1}^n \Delta_x f(x_i)[/tex]

Meaning that the result of the integral is of:

2(8 + 24 + 48 + 80) = 320 units squared.

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Answer:

Step-by-step explanation:

2n -3 =-21

2n-3+3=-21+3

2n=-19

n=-9.5

Table 1 represents a positive correlation. As the temperature rises, the number of customers also increases.

What is a correlation?

A positive correlation is a relationship between two variables in which an increase in one variable is associated with an increase in the other variable.

Table 1 represents a positive correlation. As the temperature rises, the number of customers also increases.

To plot the set of data points from Table 1, we can use a scatter plot. Each data point in the table would be represented by an (x,y) coordinate, where x is the increase in temperature and y is the number of customers. So, in this case, we would have 8 points in the scatter plot, one for each pair of data in table 1.

The plot would look like a group of points in the coordinate plane with a general upward trend, as the x-values (increase in temperature) increase, the y-values (number of customers) also increase. It's possible to observe a pattern in the points, this pattern confirms that there is a positive correlation between temperature and the number of customers.

Hence, Table 1 represents a positive correlation. As the temperature rises, the number of customers also increases.

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The exponential function that models this situation is given as follows:

y = 3(5)^x.

The y-intercept of the exponential function represents the number of people that were called by the administrators.

The graph of the exponential function is given by the image presented at the end of the answer.

The general format of an exponential function is given as follows:

y = a(b)^x.

In which:

First, the 3 school administrators decide when an announcement is needed, hence the parameter a, representing the y-intercept, is given as follows:

a = 3.

In each round, each person inserted into the conversion calls 5 more people, hence the parameter b is given as follows:

b = 5.

Meaning that the function is defined as follows:

y = 3(5)^x.

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Answer: put the dot on the positive 6 vertical line. Then put a point on the point (4,5) and the point (8,4)

Step-by-step explanation:

Answer:

86

Step-by-step explanation:

let the numbers be x,x+2,x+4,x+6

Then x+x+2+x+4+x+6=332

4x=332-12=320

x=320/4=80

so numbers are 80,82,84,86

Answer:86

Step-by-step explanation:

The order pair solution of the given system of equations is (-5, -6).

Here we will use the substitution method to solve the given equation. In this obtain value of one variable from one equation and substitute the value in another equation to get the value of variables.  

Here we have

3x - 5y = 15 ---- (1)

2x - y = - 4 ---- (2)  

=> y = 2x + 4 --- (3)

From (1) and (3)

=> 3x - 5(2x + 4) = 15

=> 3x - 10x - 20 = 15

=> -7x = 35

=> x = -5

Substitute x = - 5 in 2x - y = - 4  

=> 2(-5) - y = -4

=> -10 - y = -4

=> y = - 6

Therefore,

The order pair solution of the given system of equations is (-5, -6).    

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The total amount Derek repaid after 5 years of quarterly payments at 10% per annum is $12,829.40.

The quarterly payments represent periodic payments made every three months to offset the credit.

The periodic payments can be computed using an online finance calculator.

Where the periodic payment is given, the total payment made can be determined as the product of the periodic payments and the number of periods involved.

N (# of periods) = 20 quarters (5 years x 4)

I/Y (Interest per year) = 10%

PV (Present Value) = $10,000

PMT (Periodic Payment) = $-641.47

Results:

FV = $-0.03

Sum of all periodic payments = $-12,829.40

Total Interest = $2,829.43

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Derek borrowed $10,000 at 10% per annum compounded quarterly and repays $641.47 quarterly.  How much money did he repay after 5 years?

The given function has one root i.e. -3.

The Synthetic approach can speed up polynomial division, especially when the result needs to be divided by a linear factor. It is often used to find polynomial roots or zeroes rather than dividing components.

In algebra, synthetic division is a technique for manually dividing polynomials according to Euclid, requiring less writing and calculation than long division. Although the approach can be applied to division by any polynomial, it is often taught for division by linear monic polynomials.

We've f(x) = x³-4x+15

               = (x+3) ( x²-3x+5) + 0

which implies that it has remainder as 0, So -3 is a root of given f(x).

Since x²-3x+5 can't be factorized more, the given function has only one root i.e. -3.

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-2.25,-2.5,2.75 I think

Answer:

521621

Step-by-step explanation:

38600=7.4

x=100

*cross multiply*

100x38600/7.4

=521621.6216

nearest whole number: 521621

Answer: 28%

Step-by-step explanation: So, if we know that Mia's team won 18 of the 25 games they played, we need to subtract 18 from 25. We get 7. So, now we need to use this formula to solve:

Part percent / Whole number * 100

Let's plug in the numbers:

7 / 25 * 100

0.28 * 100

= 28

Therefore, they lost 28% of their games. I hope this helped!

Answer:

28%

Step-by-step explanation:

To find percentages, we need to know the number of games lost out of 25, then multiply it by 4 to get our percentage. We know that she won 18 of 25. 25 - 18 =7, so she lost 7 games. This gives us the fraction:

[tex]\frac{7}{25}[/tex]

Then, we will multiply this fraction by 4 to get it out of 100:

[tex]\frac{28}{100}[/tex]

This gives us 28/100, to find the percentage out of a fraction with a denominator of 100, we multiply by 100 and add a % symbol. This gives us:

[tex]\frac{28}{100} * 100 = 28%[/tex]

Then, we add our percent symbol, meaning Mia's soccer team lost 28% of their games.

Hope this helped!

The surface area of the cylinder is 791.28 sqft.

A three-dimensional object's surface area is the sum of all of its faces. Real-world applications of the concept of surface areas include wrapping, painting, and eventually building things to achieve the best possible design.

The formula for the volume of a hollow cylinder is V = (R2 -r2)h cubic units if "R" denotes the outer radius, "r" denotes the inner radius, and "h" denotes the height.

Given,

radius=9ft

height=5ft

[tex]SA=2\pi rh+2\pi r^2\\SA=2*3.14*8*5+2*3.14*5^2\\SA=791.26 sqft[/tex]

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The triangles ∆AEC ≅ ∆AFC by Side-Angle-Side theorem

The complete question is added as an attachment

With the given information and the image attached below, we can state that the two triangles are congruent by SAS,

Then go ahead to use the CPCTC to show that any of their corresponding parts are congruent as well.

The proof is given as:

Statement Reason

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