A 16​-inch board is to be cut into three pieces so that the second piece is 3 times as long as the first piece and the third piece is 4 times as long as the first piece. If x represents the length of the first​ piece, find the lengths of all three pieces.

P(A) = 11/18

P(B) = 5/18

P(C) = 1/9

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

Explanation:

Check out the diagram below. I have marked your given diagram with fractions. The red fractions 1/3, 1/3, and 1/3 are probabilities that correspond to picking the left-most set of 3 branches. Each has 1/3 chance of randomly being selected. This assumes any of the branches are equally likely.

The blue fractions 1/2 and 1/3 are the probabilities for the next set of sub-branches. For instance, the upper pair of branches each have probability 1/2 of being randomly selected, since we have two of them.

The green fractions are the result of multiplying the red and blue fractions along a particular pathway. Eg: The upper-most pathway has (1/3)*(1/2) = 1/6

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The diagram below shows that:

In summary

Side notes:

The linear function that represents the relationship is given as follows:

y = 2x + 3.

The slope-intercept definition of a linear function is given as follows:

y = mx + b.

In which:

From the table, when x increases by 2, y increases by 4, hence the slope m is found as follows:

m = 4/2

m = 2.

Hence:

y = 2x + b

When x = 5, y = 13, hence the intercept b is obtained as follows:

13 = 2(5) + b

b = 3.

Meaning that the function is given as follows:

y = 2x + 3.

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The value of the segment a for the similar triangles is equal to 6.

The triangles MNO and YXO are similar, this implies that the length OX of the smaller triangle is similar to the length ON of the larger triangle

and the length OY of the smaller triangle is similar to the length OM of the larger triangle

so;

a/(a + 18) = 4/(4 + 12)

16a = 4(a + 18) {cross multiplication}

16a = 4a + 72

16a - 4a = 72 {collect like terms}

12a = 72

a = 72/12 {divide through by 12}

a = 6.

Therefore, the value of the segment a for the similar triangles is equal to 6.

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The equation of this circle in standard form is (x + 21)² + (y + 19)² = 849.

The center of the circle is at the point (-21, -19), and its radius is √849 units.

The general form of the equation of a circle that has the same radius as the above circle is x² + y² - 50x - 30y + 1 = 0.

In Mathematics, the standard form of the equation of a circle is represented by this mathematical expression;

(x - h)² + (y - k)² = r²

Where:

In order to transform the general form of the equation of a circle to a standard form, you can either make use of completing the square method or convert the equation x² + y² + Dx + Ey + F = 0 to x² + y² + (-2h)x + (-2k)y + (h² + k² - r²) = 0.

By using the following expression x² + y² + (-2h)x + (-2k)y + (h²+k²-r²) = 0, we would determine the center (h, k) and radius of this circle as follows;

-2h = 42

h = 42/2

h = -21

-2k = 38

k = 38

k = -19

For the radius (r), we have:

-47 = h² + k² - r²

-47 = (-21)² + (-19)² - r²

r² = 849

Radius, r = √849

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The graph of a proportional relationship can be described as a graph that always starts at point zero and is always a straight line graph.

A straight line graph is defined as the type of graph that is also called a linear graph which shows a relationship between two or more quantities that uses a graphical form of representation.

There are some characteristics that shows that a graph is of proportional relationship which include the following:

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Answer:

[tex]\mbox {\large \boxed{\sqrt[12]{v^{11} }}}\\[/tex]

Step-by-step explanation:

[tex]\textrm{We are asked to simplify\:$\dfrac{\sqrt[4]{v^9}}{\sqrt[3]{v^4}}$}[/tex]

Using the fact that [tex]\sqrt[n]{x} = n^{{1/n}}[/tex]

[tex]\mbox{\large {\sqrt[4]{v^9}} =\left(v^9)^{1/4} = v^{9/4}}\\\\[/tex]

[tex]\mbox{\large {\sqrt[3]{v^4}}} = (v^4)^{1/3} = v^{4/3}}[/tex]

[tex]\mbox{\large $\dfrac{\sqrt[4]{v^9}}{\sqrt[3]{v^4}}$}[/tex] [tex]= \mbox{\large \dfrac{v^{9/4}}{v^{4/3}}}[/tex]

Using the fact that [tex]\dfrac{x^a}{x^b} = x^{a-b}[/tex]

[tex]\mbox{\large \dfrac{v^{9/4}}{v^{4/3}} = v^{9/4 - 4/3}}[/tex]

[tex]\dfrac{9}{4} - \dfrac{4}{3} = \dfrac{9 \cdot 3 - 4 \cdot 4}{12} = \dfrac{27-16}{12} = \dfrac{11}{12}[/tex]

[tex]\mbox{\large \dfrac{v^{9/4}}{v^{4/3}} = v^{9/4 - 4/3}= v^{11/12} } = \sqrt[12]{v^{11} }}[/tex]

Answer
[tex]\mbox {\large \boxed{\sqrt[12]{v^{11} }}}\\[/tex]

The solutions to the equations -³/₄x + 2¹/₄ = (¹/₂)ˣ + 1 are; -1 and 1

The functions are given as:

f(x) = -³/₄x + 2¹/₄

g(x) = (¹/₂)ˣ + 1

The solution to these two simultaneous equations will be the point where they both intersect each other.

From attached graph, both functions intersect at at two coordinates that have their x-coordinates as x = -1 and x = 1

This means that- 1 and 1 represent the solutions to f(x) = g(x) or

-³/₄x + 2¹/₄ = (¹/₂)ˣ + 1

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The required complete table for the given function f(x) = 2x - 1 is given as,
  x =  1      0    -1     2
f(x) =  1     -1     -3    3

Functions are the relationship between sets of values. e g y=f(x), for every value of x there is its exists in a set of y. x is the independent variable while Y is the dependent variable.

Given function,
f(x) = 2x - 1

To complete the table,
Put x = 1
f(1) = 2(1) - 1
f(1) = 1

Similarly
f(0) = -1
f(-1) = -3
f(2) = 3

Thus, the required complete table for the given function f(x) = 2x - 1 is given as,
  x =  1      0    -1     2
f(x) =  1     -1     -3    3

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The probability that a randomly chosen high school student who took the SAT in 2015 will have a mathematics SAT score of more than 700 points is 5.8%.

The distributed samples can be solved using the z-score formula:

We can set with mean  and standard deviation, the z-score of a measure X is given by:

[tex]Z=\frac{X-u}{sd}[/tex]

The Z-score defines how many standard deviations will be measured from the mean. After finding the Z-score, we should see the z-score table and find the p-value included with this z-score.

This p-value is the probability of the value measuring smaller than X, that is, the percentile of X. by Subtracting 1 from the p-value, we get the probability of the value of the measure is greater than X.

In this problem, we have that:

u=511 sd=120.

This is 1 subtracted by the value of Z when X = 700.

[tex]Z=\frac{X-u}{sd}\\\\Z=\frac{700-511}{120}[/tex]

Z=1.575.

Z=1.575 has a p-value of 0.942

1 - 0.942 = 0.058.

5.8% probability that a randomly chosen high school student who took the SAT in 2015 will have a mathematics SAT score of more than 700 points.

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The equation for a parabola that has x-intercepts (5,0) and (6,0) and y-intercept (0,1) is: [tex]y =[/tex] [tex]a(x + 0.5)^2 - 1[/tex]

The equation of a parabola can be found by using the vertex form of a parabolic equation, which is given by:[tex]y = a(x - h)^2 + k[/tex], where (h, k) is the vertex of the parabola. In this case, the x-intercepts (5, 0) and (-6, 0) can be used to find the value of h, which is the midpoint between the x-intercepts and is equal to (-6 + 5) / 2 = -0.5.

The y-intercept (0, -1) can be used to find the value of k. So, the equation is:[tex]y = a(x + 0.5)^2 - 1[/tex], where a is a constant that can be determined by substituting the vertex coordinates (h, k) into the equation.

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The dimensions of the rectangular region that will maximize the total area are 20ft x 50ft.

The goal is to find the dimensions of a rectangle that will maximize the total area when it is divided into 6 smaller rectangles.

Let the length of the rectangle be x and the width be y.

The total length of the fencing used to enclose the rectangle is 100 feet, so 2x + 2y = 100.

To divide the rectangle into 6 smaller rectangles, we need 5 partitions,

so we have x/6 = y, or y = x/6.

Substituting y = x/6 into 2x + 2y = 100 gives 2x + 2x/3 = 100, or x/3 = 50.

Solving for x, we find x = 150.

Then y = x/6 = 150/6 = 25.

The maximum area of the rectangle is 150 * 25 = 3750 square feet.

To maximize the area of the rectangular region, the width and length should be in the ratio of 1:2. This allows the rectangular region to be split into six smaller rectangles of equal area by placing five fences parallel to the width.

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The product  of two numbers 2.8 × 10³ and 9.7 × 10³ can be represented in scientific notation as, 2.716 x 10⁷

The standard form for a number is a method by which we write number in a format like b×10ˣ. The value of b should be in between 0 to 10.

Given that,

The numbers,

2.8 × 10³

And  9.7 × 10³

the product in scientific notation = ?

the product in scientific notation = 2.8 × 10³ x 9.7 × 10³

It is known that,

if same numbers are in multiplication form,

their powers can be added

the product in scientific notation = 27.16 x 10³⁺³

                                                       = 27.16 x 10⁶

It can be further written as,

multiplied and divide by 10

the product in scientific notation = 27.16 x 10⁶x10/10

                                                       = 2.716 x 10⁷

Hence, the product is 2.716 x 10⁷

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The combinations of items he can buy are:

1 popcorn balls and 1 candy stick.

2 candy sticks

Zeg has $9.20 left

First, calculate the price of items in his shopping basket:

2 giant lollipops = 2 * $1.75 = $3.5

3 popcorn balls = 3 * $0.5 = $1.5

1 candy bar = 1 * $0.99 = $0.99

5 candy sticks = 5 * $0.45 = $2.25

Total price =  $3.5 + $1.5 + $0.99 + $2.25 = $8.24

Amount left = $9.20 - $8.24 = $0.96

Since the price of 1 popcorn balls and 1 candy stick = $0.5  + $0.45 = $0.95

Also, 2 candy sticks = 2 * $0.45 = $0.90.

Notice that these two are still within $0.96 we have left

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The range of score Jim must get to have an average score of 94 or more is x≥ 131

inequality, In mathematics is a statement of an order relationship with greater than, greater than or equal to, less than, or less than or equal to between two numbers or algebraic expressions.

Average is the sum of items divided by the number of items.

Represent the theirs score by x

(78+73+x)/3 ≥ 94

151+x ≥ 94×3

151 + x ≥ 282

x ≥ 282-151

x≥ 131

therefore for Jim to get an average greater than 94, his third score must be greater than 131

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Answer: The function f(x) = √x + 4 describes a radical function that maps a real number x to its square root, shifted to the right by 4 units. The graph of this function is an upward-opening parabola that intersects the x-axis at x = -4.

Here are four points on the graph of the function:

The leftmost point: (0, 4)

An additional point to the right of the leftmost point: (1, 4.5)

Another additional point: (2, 5)

The third additional point: (9, 5.5)

Note: The values of x used in these examples are arbitrary and can be any real numbers. These points are only intended to give you an idea of what the graph of the function might look like

Step-by-step explanation:

Answer:

The Panthers soccer team does not like to lose because they always try to be their best. One day, they were embarrassed when all their shirts were so loose they started to fall off during a game. Guess who did not win that day?

Continuous random variable is a random variable that can take on a continuum of worth. In alternative words, a random variable is said to be continuous if it supposes a value that falls between a particular interval.

A continuous random variable can be described as a random variable that can take on an infinite number of possible values. Due to this, the probability that a continuous random variable will take on a fixed value is 0.

The probability density function of a continuous random variable can be explained as a function that gives the probability that the value of the random variable will drop between a range of values. Let X be the continuous random variable, then the formula for the pdf, f(x).

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Answer:

The width of the walkway is 5ft long

The solution of the linear equations −2x + 3y = 15 and x + y = 10 will be (3. 7). Then the correct option is B.

The allocation of weights to the important variables that produce the calculation's optimum is referred to as a direct consequence.

The system of linear equations is given below.

−2x + 3y = 15         ....1

x + y = 10

2x + 2y = 20           ...2

Add equations 1 and 2, then we have

5y = 35

y = 7

Then the value of the variable 'x' is given as,

x + 7 = 10

x = 3

The solution of the linear equations −2x + 3y = 15 and x + y = 10 will be (3. 7). Then the correct option is B.

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The value of c does not affect the covariance of Y and Z.

The covariance of Y and Z can be calculated as Cov(Y, Z) = E[YZ] - E[Y]E[Z]. Since X is Bernoulli with parameter p and E[X] = p, we have E[Z] = pE[Y]. Hence,

Cov(Y, Z) = E[YZ] - E[Y]E[Z]

= E[YXY] - pE[Y]^2

= pE[Y^2 X] - pE[Y]^2

= p(E[Y^2]E[X] + Var[Y]p) - pE[Y]^2

= p(1 * p + 1 * p^2) - p^2

= p - p^2

It can be seen that the value of c does not affect the covariance of Y and Z.

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Answer: 2621.92

Step-by-step explanation:

The addition of the numbers 2610 and 11.7 and 0.22 is 2621.92.

Hence the correct option is given by (D).

Addition is a mathematical operation to sum up between two or more than two mathematical quantities or numbers specially.

Here given the numbers are 2610 and 11.7 and 0.22.

So the addition of the numbers 11.7 and 0.22 is given by,

= 11.7 + 0.22

= 11.92

So the sum of 2610, 11.7 and 0.22 is given by,

= 2610 + 11.7 + 0.22

= 2610 + 11.92

= 2621.92

Hence the correct option is given by (D).

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The student council is selling t-shirts to increase awareness for the local animal shelter. For each t-shirt they sell, they will donate $5.00 to the shelter.

This means that the amount of money the student council will donate to the shelter will be directly proportional to the number of t-shirts they sell. The more t-shirts they sell, the more money they will donate to the shelter.

By selling t-shirts, they are taking a proactive step to help raise awareness and support for the local animal shelter.

Directly proportional: as one amount increases, another amount increases at the same rate. The symbol for "directly proportional" is ∝

(Do not confuse it with the symbol for infinity ∞)

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Accordingly, x0 = 0, x1 = 3, x2 = 0, and x3 = 3 are the first three iterations of f ( x) = -2x + 3 when x0 = 0.

A loop is described as a section of code that runs repeatedly. The procedure in which a code fragment is run just once is referred to as iteration. One iteration is the single time a loop is run. A loop may run through numerous iterations.

x1 = f(x0) = f(0) = -2 * 0 + 3 = 3

x2 = f(x1) = f(3) = -2 * 3 + 3 = 0

x3 = f(x2) = f(0) = -2 * 0 + 3 = 3

So the first 3 iterates of the function f(x) = -2x + 3 when x0 = 0 are:

x0 = 0, x1 = 3, x2 = 0, x3 = 3

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the proportionality ratio of the given Data is solved below.

the property of having suitable proportions in terms of size, number, degree, harshness, etc. If a defensive action against an unfair attack results in destruction that contravenes the proportionality criterion, it may even go far beyond a justifiable defense.

Given a figure where Two lines are intersecting three individual parallel lines.

Since Proportionality in Lengths is When quantities have the same relative size. In other words, they have the same ratio.

thus,

as we can observe the proportionality ratio for the given figures:

Therefore, the proportionality ratio of the given Data is solved above.

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2 cups of 10% sugar solution and 6 cups of 30% sugar solution.

A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements

Let,

x = number of cups of the 10% solution

y = number of cups of the 30% solution

As this is the whole amount we need, the two mixtures sum up to 8 cups. Consequently, the first equation is just x + y = 8

Solve for y to get y = 8-x. We'll use this equation later.

The second equation is a bit more tricky.

0.10x = amount of pure sugar from just the 10% batch

0.30y = amount of pure sugar from just the 30% batch

0.10x+0.30y = total amount of pure sugar from both batches

25% of 8 = 0.25×8 = 2 = total amount of pure sugar required

0.10x+0.30y = 2 is the second equation

Apply substitution to solve for x and y

0.10x+0.30y = 2 ..... start with the second equation

0.10x+0.30(8-x) = 2 ....... plugin y = 8 - x

0.10x + 2.4 - 0.30x = 2 ..... distribute

- 0.20x + 2.4 = 2 - 0.20x = 2-2.4 ....... subtract 2.4 from both sides

-0.20x = - 0.4

x = -0.4/(-0.20)

x = 2

Now that we know the value of x, we can find y.

y = 8 - x

y = 8 - 2 ... plug in x = 2

y = 6

So we need 2 cups of the 10% solution and 6 cups of the 30% solution.

As a check,

x + y = 2 + 6 = 8 works out

0.10x + 0.30y = 0.102 + 0.306 = 0.2 + 1.8 = 2 works out as well

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if you wanted to visualize the change in the mean test score for the same exam over multiple semesters line chart will be used.

A line chart connects a number of data points with a continuous line to graphically depict the historical price movement of an item. The simplest sort of chart used in finance, it often merely shows the closing prices of a security over time.

A line chart, also known as a line graph or a line plot, uses a line to link a group of data points. This type of graph uses sequential values to show trends. The x-axis (horizontal axis) often shows a succession of numbers in consecutive order.

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w, r, the coefficient of w, the constant term in the equation, the intercept parameter, equal to the slope parameter in units are ans of fill in the blank respectively.

A simple linear regression equation is used to model the relationship between two variables, where one variable is the independent or explanatory variable and the other is the dependent or response variable. The equation has the form:

y = β0 + β1x,

where y is the dependent variable, x is the explanatory variable, β0 is the intercept parameter, and β1 is the slope parameter. The intercept parameter represents the value of y when x is zero, and the slope parameter represents the change in y for each one-unit change in x. In this problem, the explanatory variable is w, and the dependent variable is r. The slope and intercept parameters need to be estimated from the data to obtain the regression equation. The equation can be used to make predictions about the response variable given a certain value of the explanatory variable.

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