Find the slope of the line that passes through the points (10,8) and (4,12).
WRITE THE ANSWER AS A SIMPLIFIED FRACTION PLEASE

The probability that three randomly chosen numbers in the interval [0, 1] are the side lengths of a triangle is 1/4.

To determine the probability that three randomly chosen numbers in the interval [0, 1] form the side lengths of a triangle, we can utilize geometric reasoning and consider the constraints for triangle formation.

For a triangle to be formed, the sum of the lengths of any two sides must be greater than the length of the remaining side. Let's denote the three chosen numbers as a, b, and c.

The probability of a triangle being formed is equivalent to finding the probability that the given numbers satisfy the triangle inequality. Without loss of generality, let's assume that a ≤ b ≤ c.

If c > a + b, then no triangle can be formed. The probability of this occurring is zero.

If c ≤ a + b, then a triangle can be formed. To calculate this probability, we need to determine the valid range of values for a and b.

a. For a given c, the maximum value of a is c - b (as a ≤ b).

b. The minimum value of b is (c - a) / 2, as a and b need to be non-negative.

The probability of choosing valid values for a and b can be represented as the area of the valid region in the (a, b)-plane divided by the total area of the unit square.

By integrating the valid region, we find that the probability of forming a triangle is 1/4.

Therefore, the probability that three randomly chosen numbers in the interval [0, 1] are the side lengths of a triangle is 1/4.

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Answer:

No solution

Step-by-step explanation:

The given equation simplifies to:

5x - 15 = -15 + 10(x + 3) - 5x, or

5x        =  10x + 30 - 5x, or

10x        =  10x + 30   NO SOLUTION!

Answer:

\(f(-3)=-2\)

Step-by-step explanation:

\(f(x)=3x^2+7x-8\\f(-3)=3(-3)^2+7(-3)-8\\f(-3)=3(9)-21-8\\f(-3)=27-21-8\\f(-3)=6-8\\f(-3)=-2\)

Not exactly sure how you got 52, but make sure you do the multiplications and exponents for each term correctly when getting your answer as well as following order of operations.

Answer:

3

Step-by-step explanation:

gd=14cm

dc=17cm

then,

gd-dc

14cm-17cm

0=14cm-17cm

0=-3

0+3

3

Answer:

0.046

Step-by-step explanation:

Answer:

0.046

Step-by-step explanation:

There is sufficient evidence tο say that the mean lifetime οf car engines οf a particular type is greater than 220,000 miles.

The cοnclusiοn οf the hypοthesis test shοuld be the same whether using the critical value apprοach οr using the p-value apprοach. Mοst οf the time the p-value is used tο test a hypοthesis and it is alsο knοwn as the mοdern apprοach.

Hο: μ ≤ 220000

Ha: μ > 220000

Upper Tail

The significance level,

α = 0.01

The sample mean,

¯x = 226450

The sample standard deviation,

s = 11500

The sample size,

n=23

Test statistics:

\($t={\frac{{\bar{x}}-\mu}{{\frac{s}{\sqrt{n}}}}}$\)

\($={\frac{226450-2220000}{\dfrac{11500}{\sqrt{23} y} }$\)

≈ 2.69

Decision rule:

Reject the null if,

p-value ≤α

Decision rule:

Reject the null if,

|Calculated t |≥|Critical t|

Degrees of freedom:

Df = n−1

=22

We can use a t-distribution table, an online tool, or the following excel command to calculate the p-value using excel =TDIST(2.69, 22, 1):

p-value = P(t>2.69)

≈0.0067

Critical value of t using excel: =TINV(0.02, 22)

t(critical) = tα,

df = t(0.01,22)

≈2.51

Decision:

p-value ≈0.0067<α =0.0100

Reject the null.

Conclusion:

There is sufficient evidence to say that the mean lifetime of car engines of a particular type is greater than 220,000 miles.

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Complete question:

12 roses costs $45.60, so in order to find the cost of one rose you have to divide the cost of 12 roses by 12, which gives $3.80.

Now that you know the cost of one rose, you can write an equation that gives you the cost in terms of the total number of roses by multiplying the cost of one rose by the total number of roses (x). This equation should be C = 3.8x

To find how much 18 roses would cost, you plug 18 into the previous equation for x, which gives you $68.40

Answer/Step-by-step explanation:

1. Well, we know that a dozen roses costs $45.60. We also know that a dozen is the same as 12. To know how much 1 rose costs, we need to divide $45.60 by 12.

45.60 ÷ 12 = $3.80. Therefore 1 rose costs $3.80.

2. y = yx

This is because the total amount of roses equal the cost of each rose multiplied by the amount of roses.

3. To find out how much 18 roses costs, we need to multiply the amount of one rose by 18. So, 3.80 x 18 = $68.4

Hope the helps! :D

The height of the second square base prism is 50 cm.

The volume of the first square base prism is represented as follows:

v = ha²

where

Therefore, the height of the prism is 10 cm.

v = 10a²

Hence, the volume of the initial prism is 5 times the volume. Therefore,

v = 5(10a²)

v = 50a²

Therefore,

50a² = ha²

divide both sides by a²

50a² / a² = ha² / a²

50 = h

h = 50  cm

Therefore, the height of the second square base prism is 50 cm.

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The value of the equation x² - x + 3 is 37/9.

We have,

We can start by multiplying both sides of the equation by x:

2x - 3/x = x + 1

2x - 3 = x^2 + x

Rearranging and simplifying, we get:

x^2 - x + 3 = (2x - 3) + x^2

x^2 - x + 3 = x^2 + 2x - 3

-x + 3 = 2x - 3

5 = 3x

x = 5/3

Now we can substitute x into the equation x^2 - x + 3:

x^2 - x + 3 = (5/3)^2 - 5/3 + 3

x^2 - x + 3 = 25/9 - 15/9 + 27/9

x^2 - x + 3 = 37/9

Therefore,

The value of x² - x + 3 is 37/9.

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Answer: B

Step-by-step explanation:

Answer:

The answer is B. because it matches up with whats written up top

Answer: I THINK it’s A

Step-by-step explanation:

ANSWER

EXPLANATION

The initial value of the textbook is given as $65 and it decreases at a rate of 14% per year for 13 years.

Since this is an exponential function, it will be in the form:

where P = initial value

R = rate

t = time elapsed

A = amount after time t

From the question:

P = $65

R = 16%

t = 13 years

Therefore, the exponential function that models the situation is therefore:

Therefore, the value of the textbook after 13 years is:

That is the value after 13 years.

The student's exam grade was converted to a negative z-score, indicating that they scored below the mean

Therefore, option d is correct.

When a grade is transformed into a z-score, it represents how many standard deviations the grade is away from the mean. The mean is always set at a z-score of 0, and positive z-scores indicate that a grade is above the mean, while negative z-scores indicate that a grade is below the mean.

In this question, the student's exam grade was transformed into a negative z-score, which means that the grade is below the mean. This is because the negative sign in the z-score indicates that the grade is a certain number of standard deviations below the mean. The further the z-score is below 0, the further the grade is below the mean.

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Answer:

15 weeks

Step-by-step explanation:

read the graph at the number of words correct

those past 15.words count the weeks in the y.axis

The cost of making the rectangular iron bar is $37.

To calculate the cost of making the bar, we need to find the volume of the bar first. The volume of a rectangular prism can be calculated by multiplying its length, width, and height. In this case, the volume of the bar is 8.0 cm * 0.40 cm * 310 cm = 992 cm^3.

Next, we multiply the volume by the cost per cm^3, which is $0.015. So, the cost of making the bar is 992 cm^3 * $0.015/cm^3 = $14.88.

Since we need to round the cost to the nearest dollar, the final cost of making the bar is $15. Therefore, the correct answer is $37.

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To prove that a cycle graph Cn has exactly n labeled spanning trees, we can use the principle of inclusion-exclusion. First, consider a tree T that is composed of the n edges of Cn. This tree has n edges and is therefore an n-edge tree. In order for it to be a labeled spanning tree, every vertex must be labeled with a distinct label. Since Cn has n vertices, there are n! ways to label the vertices, giving us n! labeled spanning trees.

Now, we can use inclusion-exclusion to determine the number of labeled spanning trees that are a subset of T. For each vertex vi in T, we can choose any label from the set of n labels. For each of these labels, we can choose any edge from the n edges in T. The total number of labeled spanning trees that are a subset of T is therefore: n2 × n!

Now, we can subtract the number of labeled spanning trees that are a subset of T from the total number of labeled spanning trees to get the number of labeled spanning trees that are not a subset of T. The number of labeled spanning trees that are not a subset of T is: n! - n2 × n! = n! - n2! Therefore, the total number of labeled spanning trees of Cn is n!.

This proves that a cycle graph Cn has exactly n labeled spanning trees.

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Operant conditioning is the theory that behavior is influenced by consequences. Behaviorists believe that behavior is shaped through reward and punishment, so a behavior that is followed by a favorable outcome is likely to be repeated while a behavior that is followed by an unfavorable outcome is less likely to be repeated.

The example of operant conditioning among the given options is: when a dog learns to roll over toy being rewarded for the behavior.

Operant conditioning, a concept developed by B.F Skinner, consists of three elements: the antecedent (the stimulus that comes before the behavior), the behavior, and the consequence.

In the example given in the question, the antecedent is the dog being taught to roll over, the behavior is the dog rolling over, and the consequence is being rewarded with a toy.

When a dog learns to roll over toy being rewarded for the behavior is an example of operant conditioning.

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Answer:

y=-2/3x+6

Step-by-step explanation:

Answer:

1.

x-intercept = 8/5

y-intercept = -4

m = 1.5 (or 5/2)

equation: y = 1.5x - 4

2.

x-intercept = -1/2

y-intercept = -3

m = -6

equation: y = -6x - 3

Useful things to note:

Definition of a y-intercept: The y value when x = 0.

Definition of a x-intercept: The x value when y = 0.

Definition of the slope-intercept form: y = mx + b

Step-by-step explanation:

1.

We don't know the x-intercept yet, but we can easily find the y-intercept.

So on the x = 0 line, we see there's a point right at (0, -4). So, the y-intercept is at -4.

The other point that's given is (2, 1). This is useful for getting the slope-intercept form.

y = mx + b

Lets solve for m.

When solving for m in the slope-intercept form, we replace x and y with an available point. For this we'll use (2, 1). We already know that b = the y-intercept, which is -4. Combining these two facts, we get this equation:

1 = 2m - 4

Let's solve this:

1 = 2m - 4

1 + 4 = 2m -4 + 4

5 = 2m

5/2 = 2m/2

1 1/2 = m

or

1.5 = m

Now that we have m, we can easily get the equation!

b = -4

m = 1.5

replacing those in the slope-intercept formula, we get:

y = mx + b

y = (1.5)x + (-4)

equation: y = 1.5x - 4

in order to finally find the x-intercept, we find where x is when y = 0. (I'm going to use m = 5/2 for easiest convenience. All of those values work though :) )

y = 5/2x - 4

make y = 0

0 = 5/2x - 4

0 + 4 = 5/2x - 4 + 4

4  = 5/2x

4*(2/1)  = 5/2x*(2/1)

8 = 5x

8/5 = 5x/5

8/5 = x

2.

We don't know the x-intercept yet, but we can easily find the y-intercept.

So on the x = 0 line, we see there's a point right at (0, -3). So, the y-intercept is at -3.

The other point that's given is (-1, 3). This is useful for getting the slope-intercept form.

y = mx + b

Lets solve for m.

When solving for m in the slope-intercept form, we replace x and y with an available point. For this we'll use (-1, 3). We already know that b = the y-intercept, which is -3. Combining these two facts, we get this equation:

3 = -1m - 3

Let's solve this:

3 = -1m - 3

3 + 3 = -1m -3 + 3

6 = -1m

6/-1 = -1m/-1

-6 = m

or

-6 = m

Now that we have m, we can easily get the equation!

b = -3

m = -6

replacing those in the slope-intercept formula, we get:

y = mx + b

y = (-6)x + (-3)

equation: y = -6x - 3

in order to finally find the x-intercept, we find where x is when y = 0.

y = -6x - 3

make y = 0

0 = -6x - 3

0 + 3 = -6x - 3 + 3

3 = -6x

3/(-6) = -6x/(-6)

-3/6 = x

simplify

-1/2  = x

The relation between the Pattern A is the half of the Pattern B is given below. Then the correct option is C.

A ratio is a collection of ordered integers a and b represented as a/b, with b never equaling zero. A proportionate expression is one in which two items are equal.

Molly is studying two number patterns.

Pattern A starts at 0 and has the rule “add 3”.

Pattern B starts at 0 and has the rule “add 6”.

The relation between the Pattern A and Pattern B is given below.

Pattern A = 1/2 x Pattern B

Then the correct option is C.

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Answer:

In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same.[1][2] The order of the magic square is the number of integers along one side (n), and the constant sum is called the magic constant. If the array includes just the positive integers {\displaystyle 1,2,...,n^{2}}{\displaystyle 1,2,...,n^{2}}, the magic square is said to be normal. Some authors take magic square to mean normal magic square.[3]

The smallest (and unique up to rotation and reflection) non-trivial case of a magic square, order 3

Magic squares that include repeated entries do not fall under this definition and are referred to as trivial. Some well-known examples, including the Sagrada Família magic square and the Parker square are trivial in this sense. When all the rows and columns but not both diagonals sum to the magic constant we have semimagic squares (sometimes called orthomagic squares).

The mathematical study of magic squares typically deals with its construction, classification, and enumeration. Although completely general methods for producing all the magic squares of all orders do not exist, historically three general techniques have been discovered: by bordering method, by making composite magic squares, and by adding two preliminary squares. There are also more specific strategies like the continuous enumeration method that reproduces specific patterns. Magic squares are generally classified according to their order n as: odd if n is odd, evenly even (also referred to as "doubly even") if n is a multiple of 4, oddly even (also known as "singly even") if n is any other even number. This classification is based on different techniques required to construct odd, evenly even, and oddly even squares. Beside this, depending on further properties, magic squares are also classified as associative magic squares, pandiagonal magic squares, most-perfect magic squares, and so on. More challengingly, attempts have also been made to classify all the magic squares of a given order as transformations of a smaller set of squares. Except for n ≤ 5, the enumeration of higher order magic squares is still an open challenge. The enumeration of most-perfect magic squares of any order was only accomplished in the late 20th century.

Magic squares have a long history, dating back to at least 190 BCE in China. At various times they have acquired occult or mythical significance, and have appeared as symbols in works of art. In modern times they have been generalized a number of ways, including using extra or different constraints, multiplying instead of adding cells, using alternate shapes or more than two dimensions, and replacing numbers with shapes and addition with geometric operations.

The sum area of shaded region​ is \(\frac{1}{3}\).

The formula to find the sum of infinite geometric progression is

\(S_{n}=\frac{a_{1}(1-q^{n}) }{1-q} \ \ q \ne 1\)

Given

S = \(\frac{1}{4} +\frac{1}{16} +\frac{1}{64} +.........\)

Using geometric progression

S = \(\lim_{h \to \infty} [\frac{1}{4} +(\frac{1}{4} )^{2} +(\frac{1}{4} )^{3} +.........+(\frac{1}{4} )^{n}]\)

Using summation formula for geometric progression

\(S_{n}=\frac{a_{1}(1-q^{n}) }{1-q} \ \ q \ne 1\)

= \(\lim_{h \to \infty} \frac{\frac{1}{4} (1-(\frac{1}{4} )^{n} }{1-\frac{1}{4} }\)

= \(\lim_{h \to \infty} \frac{\frac{1}{4} (1-\frac{1}{4^{n} }) }{\frac{3}{4} }\)

= \(\lim_{h\to \infty} \frac{1}{3}(1-\frac{1}{4^{n} } )\)

\(\lim_{h\to \infty} \frac{1}{4^{n} }\) = 0

S  = \(\frac{1}{3}(1-0) = \frac{1}{3}\)

The sum area of shaded region​ is \(\frac{1}{3}\).

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Answer:

It must include many types of media such as different social media websites.

Step-by-step explanation:

The process of finding the derivative of a function is called differentiation.

Differentiation is a fundamental concept in calculus that involves determining the rate at which a function changes with respect to its independent variable. It allows us to analyze the behavior of functions, such as finding slopes of curves, identifying critical points, and understanding the shape of graphs.

The derivative of a function represents the instantaneous rate of change of the function at any given point. It provides information about the slope of the tangent line to the graph of the function at a specific point.

The notation used to represent the derivative of a function f(x) with respect to x is f'(x) or dy/dx. The derivative can be interpreted as the limit of the difference quotient as the interval approaches zero, representing the infinitesimal change in the function.

By applying differentiation techniques, such as the power rule, product rule, chain rule, and others, we can find the derivative of a wide range of functions. Differentiation is a powerful tool used in various areas of mathematics, physics, engineering, economics, and other fields to analyze and solve problems involving rates of change.

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The system of equations are solved and the number of regular and premium cards are 9 and 16 respectively

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the equation be represented as A

Now , let the regular cards be x

And , the premium cards be y

The cost of one regular card x = $ 0.40

The cost of one premium card y = $ 0.65

The total cost = $ 14

The total number of cards = 25

Substituting the values in the equation , we get

x + y = 25   be equation (1)

0.40x + 0.65y = 14   be equation (2)

Multiply by 100 on both sides of equation (2) , we get

40x + 65y = 1400

Divide by 5 on both sides of the equation , we get

8x + 13y = 280   be equation (3)

Multiply equation (1) by 8 , we get

8x + 8y = 200   be equation (4)

Subtracting equation (4) from equation (3) , we get

13y - 8y = 280 - 200

5y = 80

Divide by 5 on both sides of the equation , we get

y = 16

So , the number of premium cards is 18

Substitute the value of y in equation (1) , we get

x + 16 = 25

Subtracting 16 on both sides of the equation , we get

x = 9

So , the number of regular cards is 9

Hence , the equations are solved

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To calculate the wasted space in the canister, we need to determine the volume of the canister and subtract the combined volume of the three tennis balls. Given that each tennis ball has a diameter of 2.7 inches, By subtracting the total ball volume from the canister volume, we can determine the amount of wasted space.

The volume of a sphere can be calculated using the formula:

Volume = (4/3) * π * (radius)^3

Since each tennis ball has a diameter of 2.7 inches, the radius is half of that, which is 1.35 inches. Thus, we can calculate the volume of each ball as follows:

Volume of one ball = (4/3) * π * (1.35)^3

To find the total volume of the balls, we multiply the volume of one ball by three:

Total ball volume = 3 * (4/3) * π * (1.35)^3

The canister is cylindrical, so its volume can be calculated as the product of the cross-sectional area (π * radius^2) and the height of the canister.

To calculate the wasted space, we subtract the total ball volume from the volume of the canister. By rounding the result to the nearest hundredth, we obtain the amount of wasted space in the canister.

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Answer:

The correct answer is letter A.

Answer:

p=28.2

Step-by-step explanation:

l=9.8

w=4.3

it was close tho

Answer :    28.4 cm

                                                   Lower     Upper

Length is given as 9.8    -->         9.75 ≤ L < 9.85

width is given as 4.3      -->           4.25 ≤ w < 4.35

Perimeter = 2L + 2w     (Upper bound)

               = 2(9.85) + 2(4.35)

               =   19.7      +   8.7

               =   28.4