Determine The Volume Of Each Rectangular Or Triangular Prism Round To The Nearest Tenth If Necessary (2024)

Answer:

The slope = 3

Explanations:

Note that:

The slope - Intercept form of the equation of a line takes the form y = mx + c

where m is the slope and

c is the intercept

The given equation is:

6x - 2y = -6

The equation can be re-written as:

2y = 6x + 6

2y / 2 = 6x/2 + 6/2

y = 3x + 3

The slope, m = 3

The intercept, c = 3

Solution

As seen from the table

For each values of the table

We define the variation from K to J

[tex]\begin{gathered} K\propto J \\ K=cJ\text{ (where c is constant of proportionality)} \end{gathered}[/tex]

When J = 2, K = -12

[tex]\begin{gathered} K=cJ \\ -12=c(2) \\ 2c=-12 \\ c=-\frac{12}{2} \\ c=-6 \end{gathered}[/tex]

Therefore, the formula connecting them will be

[tex]k=-6j[/tex]

Option B

The​ person's expectation if two of the envelopes contain ​$ 5 checks and two of the envelopes contain ​$ 35 checks is $5.

In the given question,

A person randomly selects one of four envelopes.

Each envelope contains a check that the person gets to keep.​

However, before the person can select an​ envelope, he or she must pay ​$15 to play.

We have to determine the​ person's expectation if two of the envelopes contain ​$5 checks and two of the envelopes contain ​$35 checks.

As we know that when the person have to select envelope then they have to pay $15.

Total number of envelop = 4

From the 4 envelop 2 have $5 each and 2 have $35 each.

So the probability of getting envelop of $5 = 2/4 = 1/2

Probability of getting envelop of $35 = 2/4 = 1/2

Let x be the amount a person gets after selecting the envelop.

So E(x) = $5×1/2 + $35×1/2

Taking 1/2 common on both side

E(x) = 1/2 ($5+$35)

E(x) = 1/2×$40

E(x) = $20

But he have to pay $15 before selecting the envelop.

So required expectation = $20−$15 = $5

Hence, the​ person's expectation if two of the envelopes contain ​$ 5 checks and two of the envelopes contain ​$ 35 checks is $5.

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we have

207.4÷61​

[tex]207.4\div61=\frac{207.4}{61}=\frac{2,074}{610}=\frac{1,830}{610}+\frac{244}{610}=3+\frac{244}{610}=3\frac{244}{610}[/tex]

simplify

244/610=122/305=4/10=2/5

therefore

the answer is 3 2/5

Answer:

C 5

Step-by-step explanation:

The two lines can both intersect the circle twice, and can intersect each other once, so 2 + 2 + 1 = 5

Part a

Remember that the linear equation in slope-intercept form is

y=mx+b

where

m is the slope or unit rate

b is the y-intercept or initial value

In this problem

the equation is of the form

C=m(n)+b

where

m=8.50

b=350

therefore

C=8.50n+350

Part b

A reasonable domain for n (number of cups)

Remember that the number of cups cannot be a negative number

so

the domain is the interval [0, infinite)

but a reasonable domain could be [0, 500]

Find out the range

For n=0 -----> C=350

For n=500 ----> C=8.50(500)+350=2,100 ZAR

the range is the interval [350,2,100]

Part c

calculate the cost

For n=100 cups ----> C=8.50(100)+350=1,200 ZAR

For n=200 cups ----> C=8.50(200)+350=2,050 ZAR

For n=400 cups ---> C=8.50(400)+350=3,750 ZAR

Part d

Average cost

Divide the total cost by the number of cups

For 100 cups ------> 1,200/100=12 ZAR per cup

For 200 cups ----> 2,050/200=10.25 ZAR per cup

For 400 cups ----> 3,750/400=9.38 ZAR per cup

Part e

it is better to order more cups, to reduce the initial ZAR 350 cost.

Part f

In this problem we have the ordered pairs

(200, 2150) and (400, 3750)

Find out the slope m

m=(3750-2150)/(400-200)

m=8 ZAR per cup

Find out the linear equation

C=mn+b

we have

m=8

point (200,2150)

substitute and solve for b

2150=8(200)+b

b=2150-1600

b=550

therefore

The linear equation is

C=8n+550

Part g

A reasonable domain could be [0, 600]

Find out the range

For n=0 ------> C=550

For n=600 ----> C=8(600)+550=5,350

The range is the interval [550,5350]

Part h

The gradient is the same as the slope

so

slope=8

that means ----> the cost of each cup is 8 ZAR

Part i

For n=600

C=8(600)+550=5,350 ZAR

Part j

we have the inequality

8n+550 < 8.50n+350

Solve for x

550-350 < 8.50n-8n

200 < 0.50n

400 < n

Rewrite

n > 400

For orders more than 400 cups is more effective to order from Cupomatic

Verify

For n=401

C=8n+550=8(401)+550=3,758 ZAR

C=8.50n+350=8.5(401)+350=3,758.5 ZAR

the cost is less in CUPOMATIC, is ok

the answer is

For orders more than 400 cups is more effective to order from Cupomatic

we know that

The first step to find out the inverse of the function is to exchange the variables (x for y and y for x)

therefore

the answer is the second option

Using the Borda's method, when one person is ranked as 1st, he/she gets 3 points, if he/she is ranked 2nd, get 2 points, also, if he/she is ranked as 3rd get 1 point, and finally, 0 points if she/he is ranked as 4th

so, let's detemine how many points got each one

Cardona: Was selected 1st by 6 people, 2nd by 2 people, 3rd by 4 people and 4th by 2 people

[tex]C=3*6+2*2+1*4=26[/tex]

So, that's a total of 26 points

Pitts-Jones: Was selected as: #1 by 4 people, #2 by 3 people, #3 by 6 people and 4th by 1 person

[tex]P=3*4+2*3+1*6=24[/tex]

So, that's 24 points for Pitts-Jones,

De Plata: Was ranked #1 by 2 people, #2 by 8 people, #3 by 1 person and #4 by 3 people

[tex]D=3*2+2*8+1*1=23[/tex]

That's 23 points for De Plata

Vincent: Was ranked as #1 by 2 people, #2 by 1 person, #3 by 3 people and #4 by 8 people

[tex]V=3*2+2*1+1*3=11[/tex]

that's 11 points for Vincent,

Answer: From the above, we can conclude that the winner using Borda's method is Cardona

To order these numbers, we begin with the whole part of each number. In the case of having two numbers with equal whole part, we look for the greatest tenth. So, the order would be

[tex]3.56;4.03;4.2;12[/tex]

Notice that, 4.03 is less than 4.2, because its tenth is less.

Answer:

[tex]l=\frac{S-\pi r^{2}}{\pi r}[/tex]

Explanation:

The surface area of a cone is calculated using the formula:

[tex]S=πr^2+πrl[/tex]

We want to solve for l.

First, subtract πr² from both sides of the equation:

[tex]\begin{gathered} S-\pi r^2=\pi r^2-\pi r^2+\pi rl \\ S-\pi r^2=\pi rl \end{gathered}[/tex]

Next, divide both sides by πr:

[tex]\begin{gathered} \frac{S-\pi r^2}{\pi r}=\frac{\pi rl}{\pi r} \\ l=\frac{S-\pi r^{2}}{\pi r} \end{gathered}[/tex]

The equation solved for l is:

[tex]l=\frac{S-\pi r^{2}}{\pi r}[/tex]

We know that, for a parabola, the minimum, or the maximum, is given by the vertex of the parabola. The formula for the vertex of the parabola is given by:

[tex]x_v=-\frac{b}{2a},y_v=c-\frac{b^2}{4a}[/tex]

And we have the coordinates for x and y for the vertex.

We can see that the line of symmetry is x = -1, and this is the same value for the value of the vertex for x-coordinate, that is, the x-coordinate is equal to x = -1.

With this value for x, we can find the y-coordinate using the given equation of the parabola:

[tex]f(x)=(x-2)\cdot(x+4)\Rightarrow f(-1)=(-1-2)\cdot(-1+4)\Rightarrow f(-1)=(-3)\cdot(3)[/tex]

We can also expand these two factors, and we will get the same result:

[tex]f(x)=(x-2)\cdot(x+4)=x^2+2x-8=(-1)^2+2\cdot(-1)-8=1-2-8=-1-8=-9[/tex]

Therefore, the value for the y-coordinate (the value for the y-coordinate of the parabola, which is, at the same time, the minimum point for y of the parabola) is:

[tex]f(-1)=(-3)\cdot(3)\Rightarrow f(-1)=-9[/tex]

The minimum point of the parabola is (-1, -9) (answer), and we used the given function (rule) to find the value of the y-coordinate.

We can check these two values using the formula for the vertex of the parabola as follows:

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

Then, a = 1 (it is positive so the parabola has a minimum), b = 2, and c = -8.

Hence, we have (for the value of the x-coordinate, which is, at the same time, the value for the axis of symmetry in this case):

[tex]x_v=-\frac{2}{2\cdot1}\Rightarrow x_v=-1[/tex]

And for the value of the y-coordinate, we have:

[tex]y_v=c-\frac{b^2}{4a}\Rightarrow y_v=-8-\frac{2^2}{4\cdot1}=-8-\frac{4}{4}=-8-1\Rightarrow y_v=-9[/tex]

Answer:

Explanation:

Part 1:4(5c + 3) and 9c + 7

[tex]4\mleft(5c+3\mright)=20c+12\neq9c+7[/tex]

The answer is NO.

Part 2: 10f – 10 and 2(8f - 5)

[tex]2\mleft(8f-5\mright)=16f-10\neq10f-10[/tex]

The answer is NO.

Part 3: 12g + 21 and 3(4g + 7)

Part 4: 6(4) – 6) and 24 - 36

The given is a discrete random variable.

For a discrete random variable, the expected value is calculated by summing the product of the value of the random variable and its associated probability, taken over all of the values of the random variable.

It is given that the probability of a flood is 1%=0.01.

It follows that the probability of no flood is (100-1)%=99%.

Hence, the expected net gain is:

[tex]E(X)=0.01(-9500)+0.99(500)=-95+495=400[/tex]

Hence, the expected net gain is $400.

The expected net gain is E(X) = $400.

Answer

Expected number of paintings that a randomly chosen exhibit has = 40.3

Explanation

The expected value of any distribution is calculated as the mean of that distribution.

The mean is the average of the distribution. It is obtained mathematically as the sum of variables divided by the number of variables.

Mean = (Σx)/N

x = each variable

Σx = Sum of the variables

N = number of variables

Σx = (9 × 2) + (21 × 1) + (40 × 1) + (46 × 3) + (52 × 1) + (67 × 2)

Σx = 18 + 21 + 40 + 138 + 52 + 134

Σx = 403

N = 2 + 1 + 1 + 3 + 1 + 2 = 10

Mean = (Σx)/N

Mean = (403/10) = 40.3

Hope this Helps!!!

The given system of equations are

y=-3x/2 + 6

y=5x - 7

We would substitute values for x into the equations and find the corresponding y values. These values would be plotted on a graph. Where the lines of both equations meet would represent the solution of the system of equations.

For the first equation,

y = - 3x/2 + 6

if x = 0, y = 3 * 0/2 + 6 = 6

If x = 1, y = - 3 * 1/2 + 6 = 4.5

if x = 2, y = - 3 * 2/2 + 6 = 3

We would plot these values on the graph

For the second equation,

y = 5x - 7

if x = 0, y = 5 * 0 - 7 = - 7

If x = 1, y = 5 * 1 - 7 = - 2

if x = 2, y = 5 * 2 - 7 = 3

We would plot these values on the graph

The diagram of the graph is shown below

Looking at the graph, at the point where both lines meet,

x = 2, y = 3

Thus, the solution is (2,3)

Answer:

C. The graph of f(x) is shifted 17 units up.

Step-by-step explanation:

When + is outside the equation, it means up.

P(B) = 2/3

P(An B) = 1/6

What will P(A) have to be for A and B to be independent?

Remember that

Events A and B are independent if the equation P(A∩B) = P(A) · P(B) holds true

substitute given values

1/6=P(A)*(2/3)

solve for P(A)

P(A)=1/4

Answer:

m = 2 and m = 3

Explanation:

To find the solutions to the inequality, we need to isolate m. So, we can subtract 7 from both sides as:

7 + m < 20

7 + m - 7 < 20 - 7

m < 13

Therefore, any number that is less than 13, is a solution of the inequality.

For example: 2 and 3 are solutions of the inequality.

Explanation

We are required to solve each equation till we arrive at the one that satisfies the "x=12" question.

First equation:

[tex]\begin{gathered} x-12=12 \\ Collect\text{ like terms} \\ x=12+12 \\ x=24 \end{gathered}[/tex]

Second equation:

[tex]\begin{gathered} x-24=12 \\ Collect\text{ like terms} \\ x=12+24 \\ x=36 \end{gathered}[/tex]

Third equation:

[tex]\begin{gathered} x-14=2 \\ Collect\text{ like terms} \\ x=2+14 \\ x=16 \end{gathered}[/tex]

Last equation:

[tex]\begin{gathered} x-5=7 \\ Collect\text{ like terms} \\ x=7+5 \\ x=12 \end{gathered}[/tex]

Hence, the last equation is the solution.

She bought pencils for her class. She bought 25 pencils for $2.50 . The amount for each pencil can be computed below

[tex]\begin{gathered} 25\text{ pencils = \$2.50} \\ 1\text{ pencil = ?} \\ \text{cross multiply} \\ \cos t\text{ of each pencil=}\frac{2.50}{25} \\ \text{ cost of each pencil = \$}0.1 \end{gathered}[/tex]

Given:

The system of equation is given as,

[tex]\begin{gathered} 7x-4y=28 \\ 5x-2y=17 \end{gathered}[/tex]

The objective is identify the augmented matrix for the system of equations and the solution using row operations.

Explanation:

The required augmented matrix will be,

Performing the Gauss-Jordan elimination with the following operation,

[tex]R_2=R_2-\frac{5R_1}{7}[/tex]

By applying the operation to the augmented matrix,

To find y :

On equating the second row of the matrix,

[tex]\begin{gathered} \frac{6y}{7}=-3 \\ y=\frac{-3}{\frac{6}{7}} \\ y=\frac{-3\times7}{6} \\ y=\frac{-7}{2} \end{gathered}[/tex]

To find x :

On equating the first row of the matrix,

[tex]\begin{gathered} 7x-4y=28 \\ 7x=28+4y \\ x=\frac{28+4y}{7} \end{gathered}[/tex]

Substitute the value of y in the above equation.

[tex]\begin{gathered} x=\frac{28+4(\frac{-7}{2})}{7} \\ x=\frac{28-14}{7} \\ x=\frac{14}{7} \\ x=2 \end{gathered}[/tex]

Thus the value of solutions are,

[tex]\begin{gathered} x=2 \\ y=-\frac{7}{2}=-3.5 \end{gathered}[/tex]

Hence, option (3) is the correct answer.

Answer:

C. 84

Explanation:

First, we will calculate the percentage of left-handed students in the group of 60 that the teacher asked.

So, the percentage is equal to:

[tex]\frac{18}{60}\times100\text{ \% = 30\%}[/tex]

Therefore, 30% of the randomly selected students are left-handed.

Now, we can use this percentage to estimate the number of left-handed students in the group of 280.

Then, 30% of 280 is equal to:

[tex]30\text{ \% }\times280=\frac{30}{100}\times280=84[/tex]

So, the answer is C. 84.

The average rate of change is:

[tex]\frac{f(-1)-f(-4)}{-1+4}=\frac{f(-1)-f(-4)}{3}[/tex][tex]f(-1)=2(-1^2)+4=6[/tex][tex]f(-4)=2(-4^2)+4=2(16)+4=36[/tex]

then computing the first formula, the average rate of change of f(x) is

[tex]\frac{6-36}{3}=-10[/tex]

The mean and the median for each data-set are given as follows:

5-th grade students:

Mean: 4.67Median: 5 hours.

7-th grade students:

Mean: 3.46 hours.Median: 4 hours.

Dot plot

A dot plot shows the number of times that each observation appears on a data-set.

Hence the hours of the 5th-graders are as follows:

1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 8, 8

The mean is the sum of all the numbers of hours divided by the number of students, hence:

Mean = (2 x 1 + 3 x 2 + 1 x 3 + 4 x 4 + 5 x 5 + 5 x 6 + 2 x 7 + 2 x 8)/(2 + 3 + 1 + 4 + 5 + 5 + 2 + 2) = 4.67.

There are 24 elements in the data-set, hence the median is the mean of the 12th and the 13th element, as follows:

Median = (5 + 5)/2 = 5.

Hence the hours of the 7th-graders are as follows:

0,0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7.

Hence the mean is:

Mean = (2 x 0 + 2 x 1 + 3 x 2 + 5 x 3 + 5 x 4 + 3 x 5 + 3 x 6 + 1 x 7)/24 = 3.46.

The 12th element is of 3, the 13th of 5, hence the median is:

Median = (3 + 5)/2 = 4.

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Given the equation:

[tex]\sqrt[]{x}+1=x-\sqrt[]{x}-1[/tex]

Solving for x:

[tex]\begin{gathered} \sqrt[]{x}+\sqrt[]{x}=x-1-1 \\ 2\sqrt[]{x}=x-2 \end{gathered}[/tex]

Now, we take the square on both sides of the equation:

[tex]\begin{gathered} 4x=x^2-4x+4 \\ 0=x^2-8x+4 \end{gathered}[/tex]

Now, using the general solution of quadratic equations:

[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]

From the problem, we identify:

[tex]\begin{gathered} a=1 \\ b=-8 \\ c=4 \end{gathered}[/tex]

Then, the solutions are:

[tex]\begin{gathered} x=\frac{-(-8)\pm\sqrt[]{(-8)^2-4\cdot1\cdot4}}{2\cdot1}=\frac{8\pm\sqrt[]{64-16}}{2} \\ x=\frac{8\pm4\sqrt[]{3}}{2}=4\pm2\sqrt[]{3} \end{gathered}[/tex]

But the original equation √(x), so x can not be negative if we want a real equation. Then, the only real solution of the equation is:

[tex]x=4+2\sqrt[]{3}[/tex]

The domain is all the values of the independent variable (in this case, x) for which the function is defined.

In this case, as it is indicated with the arrows in both ends, the function continues for greater and smaller values of x.

As there is no indication that for some value or interval of x the function is not defined (a discontinuity, for example), then it is assumed that the function domain is all the real values.

Example function:

We have the function y=1/(x-2)

We can look if there is some value of x that makes the function not defined.

The only value of x where f(x) is not defined is x=2. When x approximates to 2, the value of the function gets bigger or smaller whether we are approaching from the right or from the left.

Then, the function is not defined for x=2. So, the domain of f(x) is all the real numbers different from x=2.

The domain is, by default, all the real numbers, but we have to exclude all the values of x (or intervals, in some cases like the square roots) for which f(x) is not defined.

Hello there. To solve this question, we have to remember some properties about polynomial functions.

Given the polynomial function

[tex]p(x)=3(x+1)(x-2)(2x-5)[/tex]

We want to determine:

a) What are the x-intercepts of the graph of p(x)?

For this, we have to determine the roots of the polynomial function p(x). In this case, we have to determine for which values of x we have

[tex]p(x)=0[/tex]

Since p(x) is written in canonical form, we find that

[tex]p(x)=3(x+1)(x-2)(2x-5)=0[/tex]

A product is equal to zero if at least one of its factors is equal to zero, hence

[tex]x+1=0\text{ or }x-2=0\text{ or }2x-5=0[/tex]

Solving the equations, we find that

[tex]x=-1\text{ or }x=2\text{ or }x=\dfrac{5}{2}[/tex]

Are the solutions of the polynomial equation and therefore the x-intercepts of p(x).

b) What is the end-behavior of p(x) as x goes to +∞ or x goes to -∞?

For this, we have to take the limit of the function.

In general, for polynomial functions, those limits are either equal to ∞ or -∞, depending on the degree of the polynomial and the leading coefficient.

For example, a second degree polynomial function with positive leading coefficient is a parabola concave up and both limits for the function as x goes to ∞ or x goes to -∞ is equal to ∞.

On the other hand, an odd degree function usually has an odd number of factors (the number of x-intercepts in the complex plane) hence the limits might be different.

In this case, we have a third degree polynomial equation and we find that, as the leading coefficient is positive and all the other factors are monoic, that

[tex]\begin{gathered} \lim_{x\to\infty}p(x)=\infty \\ \\ \lim_{x\to-\infty}p(x)=-\infty \end{gathered}[/tex]

That is, it gets larger and larger when x is increasing arbitrarily, while it get smaller and smaller as x is decreasing.

c) To find the equation for a polynomial q(x) that has x-intercepts at -2, 3/4 and 7.

The canonical form of a polynomial of degree n with x-intercepts at x1, x2, ..., xn and leading coefficient equals a is written as

[tex]f(x)=a\cdot(x-x_1)(x-x_2)\cdots(x-x_n)[/tex]

So in this case, there are infinitely many polynomials satisfying this condition. Choosing a = 1, we find that q(x) is equal to

[tex]\begin{gathered} q(x)=(x-(-2))\cdot\left(x-\dfrac{3}{4}\right)\cdot(x-7) \\ \\ \boxed{q(x)=(x+2)\cdot\left(x-\dfrac{3}{4}\right)\cdot(x-7)} \end{gathered}[/tex]

These are the answers to this question.

Based on the given points from the scatter plot on the number of CDs sold in millions, the line of best fit for the data is y = -45x + 90,895

The estimated number of CDs sold in 2008 was 535 CDs.

How to find the line of best fit?

The line of best fit will take the form:
y = Slope(x) + y intercept

The value of x will be assumed to be the number of years since 1999.

The slope is:

= Change in y / Change in x

= (805 - 940) / (2002 - 1999)

= -45

The y intercept is:
940 = -45(3) + y

y = 90,895

The line of best fit is:

y = -45x + 90,895

This means that the number of CDs sold in 2008:

= -45(2008) + 90,895

= 535 CDs

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Option C

Points: A, B, C, D

Line segments: AB, AD, AC, DC, BC