The box plots display measures from data collected when 20 people were asked about their wait time at a drive-thru restaurant window.

A horizontal line starting at 0, with tick marks every one-half unit up to 32. The line is labeled Wait Time In Minutes. The box extends from 8.5 to 15.5 on the number line. A line in the box is at 12. The lines outside the box end at 3 and 27. The graph is titled Super Fast Food.

A horizontal line starting at 0, with tick marks every one-half unit up to 32. The line is labeled Wait Time In Minutes. The box extends from 9.5 to 24 on the number line. A line in the box is at 15.5. The lines outside the box end at 2 and 30. The graph is titled Burger Quick.

Which drive-thru is able to estimate their wait time more consistently and why?

Burger Quick, because it has a smaller IQR
Burger Quick, because it has a smaller range
Super Fast Food, because it has a smaller IQR
Super Fast Food, because it has a smaller range

If the price of Aloysius' favorite pecan pie has gone up 225% in one year and the pie now costs $26, the cost one year ago was $11.55.

The old price of the pie can be determined by proportions.

Proportion refers to the equation of two ratios.

The current price of the pecan pie (in percentage terms) = 225%

The current price or cost of the pie = $26

Proportionately, 225% = $26, and the old price a year ago = 100%.

The old price a year ago of the pie = $11.55 ($26 × 100 ÷ 225)

Thus, if Aloysius bought his favorite pecan pie last year, the cost would have been $11.55.

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The area left for development in the field outside the playground is C)1024[tex]ft^2[/tex].

The region that an object's shape defines as its area. The area of a figure or any other two-dimensional geometric shape in a plane is how much space it occupies.

We know  that area of rectangle formula is,

Area = [tex]length\times breadth[/tex] square unit

The total area of the field is the area of the large rectangle, which is:

=> [tex]28 \times48 = 1,344 ft^2[/tex]

The area of the playground is the area of the smaller rectangle, which is:

=> [tex]20 \times 16 = 320 ft^2[/tex]

Therefore, the area left for development outside the playground is:

=> [tex]1,344 ft^2 - 320 ft^2 = 1,024 ft^2[/tex]

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Answer:

Answer below :)

Step-by-step explanation:

To determine which bus is the most consistent, we need to look at the measures of variability for each bus. The two measures of variability that are commonly used are the range and the interquartile range (IQR).

The range is the difference between the maximum and minimum values in a dataset. It gives an idea of how spread out the data is but can be affected by extreme values or outliers. The IQR is a better measure of variability as it is not affected by extreme values and is a more robust measure of variability.

Looking at the given data, Bus 47 has a range of 8 and an IQR of 8, while Bus 14 has a range of 6 and an IQR of 6. This indicates that Bus 14 has less variability in its travel times than Bus 47.

Therefore, we can conclude that Bus 14 is the most consistent in terms of travel times.

The length of EC is 8 and the length of CD is 16

How to solve the question?

Pythagoras theorem is a fundamental theorem in mathematics named after the ancient Greek mathematician Pythagoras. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs).

The theorem can be expressed mathematically as:

c^2 = a^2 + b^2

where c is the length of the hypotenuse and a and b are the lengths of the other two sides. This means that if we know the lengths of any two sides of a right-angled triangle, we can use Pythagoras theorem to find the length of the third side.

The theorem has many practical applications, including in construction, engineering, and physics. It is also a key concept in trigonometry and is used extensively in various fields of science and mathematics.

We can use the Pythagorean theorem to solve this problem. According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. In this case, we know that the length of the hypotenuse (c) is 10 and the length of one leg (a) (the base) is 6. We can use this information to find the length of the other leg (b) (the perpendicular) as follows:

c² = a²+ b²

10²= 6² + b²

100 = 36 + b²

b²= 64

b = 8

Therefore, the length of the perpendicular (EC) is 8 units.

for CD it is given in question that EC=ED  there fore

ED=8

CD=EC+ED

CD=8+8

CD=16

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Probability that a shipment will be returned if the true sample proportion of defective cartridges in the shipment is 0.06 is equals to the 99.4457%.

We have A manufacturer of computer printers purchases plastic ink cartridges from a vendor.

Sample size for cartridge sample, n

= 238

Sample proportion of defective cartridges is more than 0.02.

true proportion of defective cartridges in the shipment = 0.06

Population Mean, μ = Σ(x) = 0.06 × 238

= 14.28

standard deviations, σ = sqrt(V(x))

= sqrt(238×0.06×0.94) = 3.74

If there are more than 0.02× 238 = 4.76 defective, the sample will be returned. This probability is 1 subtracted by the pvalue of Z when x = 4.8

Using Z- score in normal distribution formula, z = (x - μ) / σ

=> z = (4.8 - 14.3) / 3.74 = -2.54

=> P(Z < -2. 54) = 0.00554

This means that there is a 1 - 0.00554

= 0.994457376556917.

Hence, required probability is 99.4457%.

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Complete question:

A manufacturer of computer printers purchases plastic ink cartridges from a vendor. When a large shipment is received, a random sample of 230 cartridges is selected, and each cartridge is inspected. If the sample proportion of defective cartridges is more than 0.02, the entire shipment is returned to the vendor. (a) What is the approximate probability that a shipment will be returned if the true proportion of defective cartridges in the shipment is 0.06?

Answer:

21

Step-by-step explanation:

Answer:

7x7=59

Step-by-step explanation:

one shakes 7 hands and there is 7 so 7 time 7

the set of factors that can be used to rewrite the expression is    

{12xy, (y + 2)}.

To rewrite the expression 12xy2 + 24xy in factored form, we need to find the greatest common factor (GCF) of the two terms, which is 12xy. We can then factor out this GCF to obtain:

12xy2 + 24xy = 12xy(y + 2)

Therefore, the set of factors that can be used to rewrite the expression is {12xy, (y + 2)}.

To understand why this set of factors is correct, we need to review some key concepts of factoring. When we factor an expression, we are essentially breaking it down into its constituent parts, which are multiplied together to give the original expression. In other words, we are looking for the factors that, when multiplied, give the original expression.

One common method of factoring is to use the distributive property of multiplication, which states that a(b + c) = ab + ac. This means that we can factor out a common factor from two or more terms by distributing it to each term. For example, if we have the expression 3x + 6, we can factor out the GCF of 3 to obtain:

3x + 6 = 3(x + 2)

Another key concept in factoring is the notion of a perfect square trinomial, which is a quadratic expression of the form a2 + 2ab + b2, where a and b are constants. This expression can be factored as (a + b)2. For example, the expression x2 + 4x + 4 is a perfect square trinomial, which can be factored as (x + 2)2.

Returning to the expression 12xy2 + 24xy, we can see that the GCF of the two terms is 12xy, since this is the largest factor that divides evenly into both terms. We can then use the distributive property to factor out this common factor:

12xy2 + 24xy = 12xy(y + 2)

This expression is now in factored form, since it consists of the product of the GCF 12xy and the binomial (y + 2). Note that the binomial (y + 2) is not a perfect square trinomial, since it is not of the form a2 + 2ab + b2. Therefore, we cannot further factor this expression using the methods of perfect square trinomials or other common factoring techniques.

In summary, to rewrite the expression 12xy2 + 24xy in factored form, we need to identify the GCF of the two terms, which is 12xy. We can then factor out this common factor using the distributive property, resulting in the factored form 12xy(y + 2). The set of factors that can be used to rewrite this expression is {12xy, (y + 2)}.

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The expression that shows Number of vowels in 25 trials is:

(3/8) * 25

Number of vowels in 25 trials is approximately 9 vowels

Number of consonants in his next 30 trials is 19 consonants

The parameters given are:

Number of tiles = 100

Letter of tiles in results = Q, S, A, B, E, S, E, M

Total number of tiles in result = 8

Total vowels = 3

The expression that shows Number of vowels in 25 trials is:

(3/8) * 25 ≅ 9 vowels

Number of consonants in his next 30 trials = (5/8) * 30 = 150/8 ≅ 19

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Answer:

D) 5h+45

Brainly asked me to put at least 20 words so I'm adding this here.

Answer: The expression equivalent to 5(h+9) is 5h+45, so the answer is option 4.

Answer: it would take 5 weeks

If the debt is normally distributed with a population standard deviation of $1100 and 15% of college seniors owe less than the amount of money is equals to the $2121.96.

The area under the standard normal curve represents to probability. The total area under the curve is equals to one. A Standard Normal Cumulative Probability, is a table which provides the cumulative probability of the left tail, as in the values less than the z-score in question. Here,

population mean, μ = $3262

standard deviation, σ = 1100

P- value = 15%

Using the normal distribution table, Z-score value is equals to - 1.0364. Now, we can use Z-scores formula is written [tex]Z = \frac{X - \mu}{\sigma }[/tex]

Substitutes the known values in above formula, - 1.0364 = (X - 3262 )/1100

=> X - 3262 = 1100× ( - 1.0364)

=> X = 3262 - 1140.04

=> X = 2121.96

Hence, required value is $ 2121.96.

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We can solve this problem using trigonometry and the properties of triangles.

Let C be the location of the radio transmitter. Then, ACB is a triangle with sides AC = x (the distance from A to the transmitter), BC = y (the distance from B to the transmitter), and AB = 2.00 mi.

We can use the fact that the sum of the interior angles of a triangle is 180 degrees to find the angle at C:

angle ACB = 180 degrees - angle BCA - angle CAB

From the information given in the problem, we know that:

angle CAB = N 36° 20' E

angle BCA = N 43° 40' W

To add or subtract angles, we need to convert them to a common direction. We can do this by adding or subtracting 180 degrees, or by using the fact that 1 degree = 60 minutes (') and 1 minute = 60 seconds ("). Therefore:

angle CAB = 36 degrees + 20/60 degrees = 36.3333... degrees

angle BCA = 180 degrees - (43 degrees + 40/60 degrees) = 136.6666... degrees

Substituting these values into the equation for angle ACB, we get:

angle ACB = 180 degrees - 136.6666... degrees - 36.3333... degrees = 7.0000... degrees

Now, using the law of sines, we can write:

x / sin(angle CAB) = 2.00 mi / sin(angle ACB)

y / sin(angle BCA) = 2.00 mi / sin(angle ACB)

Solving for x and y, we get:

x = 2.00 mi * sin(angle CAB) / sin(angle ACB) = 2.00 mi * sin(36.3333... degrees) / sin(7.0000... degrees) = 9.0734... mi

y = 2.00 mi * sin(angle BCA) / sin(angle ACB) = 2.00 mi * sin(136.6666... degrees) / sin(7.0000... degrees) = 1.1878... mi

Therefore, the distance of the transmitter from B is y = 1.1878... mi (rounded to 4 decimal places).

Two expressions that are equivalent to 7(t + 5) are: 7t + 35 and 7t + (5 * 7)

what is algebra?

Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas.

When we have an expression of the form a(b + c), we can simplify it using the distributive property of multiplication over addition. This property allows us to distribute the multiplication of a across the terms of the sum (b + c), and then add the resulting products.

In the case of 7(t + 5), we can use the distributive property to distribute the multiplication of 7 across t and 5, resulting in the expression 7t + 7(5). We can then simplify 7(5) to 35, giving us the equivalent expression 7t + 35.

Alternatively, we can simplify 7(t + 5) by first multiplying 7 and 5 to get 35, and then adding it to 7t, resulting in the expression 7t + 35.

Therefore, two expressions that are equivalent to 7(t + 5) are:

7t + 35

7t + (5 * 7)

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-68/1 is  in the form of rational number.

What is an example of an integer?

A whole number that can be positive, negative, or zero is called an integer. It is not a fraction. Examples of numbers include: -5, 1, 5, 8, 97, and 3,043. The following numbers are examples of non-integers: -1.43, 1 3/4, 3.14,.09, and 5,643. 1. A positive integer and a negative integer cannot be multiplied.

                            Two positive integers can be multiplied to make a positive number. As an integer is only a collection of numbers, there is no specific formula for it. When doing mathematical operations on numbers, such as addition, subtraction, etc., there are nonetheless specific guidelines that must be followed.

-68 as a quotient of integers

In form of rational number

 - 68 = -68/1

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The sοlutiοn οf the given equatiοn is x=7 and cοrrect.

The equal sign ('=') cοnnects twο expressiοns tο fοrm a mathematical statement. There needs tο be at least οne unknοwable variable fοr the result tο be determined. An example οf an equatiοn is 3x - 8 = 16. When this equatiοn is sοlved, the result is x = 8.

Part A:

Given equatiοn is

3x+2(4+6x)= 113----------(1)

Multiplying the bracket term by 2 we get

3x+8+12x=113

Arranging the similar terms in the left hand side we get,

3x+12x+8=113

Subtracting 8 frοm bοth sides we get,

3x+12x+8-8=113-8

⇒(3x+12x)= 105

Adding the similar terms in the left hand side we get,

15x = 105

Dividing bοth sides by x= 105/15=7

Sο sοlving the equatiοn we get x=7.

Part B:

Checking fοr the sοlutiοn:

If the sοlutiοn is right then putting the value οf x in the left hand side οf the  given equatiοn will prοduce the right hand side.

putting x=7 in the left hand side οf equatiοn (1),

3x+2(4+6x)

= 3×7+2(4+6×7)

= 21+2(4+42)

= 21+ 2×46

= 21+ 92

= 113

Which is the right hand side οf equatiοn (1).

Sο, the sοlutiοn is cοrrect and checked.

Hence, the sοlutiοn οf the given equatiοn is x=7 and cοrrect.

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Answer:

Let's call the total number of chairs in the theater "x". Since Alison's seat is 3rd from the front, there are 2 chairs in front of her. Similarly, since her seat is 18th from the back, there are 17 chairs behind her. Therefore, the number of chairs between the 2 chairs that Alison and Naida are sitting on is: x - (2 + 17) = x - 19 Now, if Naida can see 8 chairs to her left and 11 chairs to her right, that means there are 8 + 1 + 11 = 20 chairs between her and Alison. So we can set up an equation: (x - 19) - 20 = Alison's seat number Simplifying this equation: x - 39 = Alison's seat number We

The mean of the sampling distribution of the sample means is 55 and the standard deviation is approximately 1.79 (rounded to one decimal place).

Standard deviation is a measure of the amount of variation or dispersion of a set of data values from its mean or expected value. It is a statistic that represents how spread out the data is from the mean.

The standard deviation is calculated by taking the square root of the variance, which is the average of the squared differences of each data point from the mean. The standard deviation is expressed in the same units as the data and is usually denoted by the symbol "σ" (sigma) for a population or "s" for a sample.

In the given question,

We can start by using the properties of the mean and standard deviation of a sampling distribution:

The mean of the sampling distribution of the sample means (Mx) is equal to the population mean (μ), which is given as 55.

The standard deviation of the sampling distribution of the sample means (Ox) is equal to the population standard deviation (σ) divided by the square root of the sample size (n), i.e.,

Ox = σ / sqrt(n)

where n = 5 (the sample size) and σ = 4 (the population standard deviation).

Substituting these values into the formula, we get:

Ox = 4 / sqrt(5) ≈ 1.79

Therefore, the mean of the sampling distribution of the sample means is 55 and the standard deviation is approximately 1.79 (rounded to one decimal place).

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The remainder when f(x) is divided by x-1 can be calculated using the division theorem.

The remainder when f(x) is divided by x-1 is 6.

remainder is 6 and divisor is x-1.

According to the division theorem, when a polynomial function is divided by another polynomial function, the remainder is equal to the function evaluated at the point where the divisor of the division equation is equal to zero.

In this case, the divisor is x-1, and when x-1 is equal to zero, then x = 1.

Therefore, the remainder when f(x) is divided by x-1 can be calculated by evaluating the function f(x) at x = 1:

[tex]f(1) = 2(1)^3 + (1)^2 + 3[/tex]

= 2 + 1 + 3

= 6

f(1) = 2(1)³ + (1)² + 3 = 6

Therefore, the remainder when f(x) is divided by x-1 is 6.

remainder is 6 and divisor is x-1.

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Answer:

9x + 6 = 8

Step-by-step explanation:

"Sum" indicates addition problem. 9 times a number can be represented as 9x (with 9 and 'a number' being x). Then you add 6 because it states 'and 6'. Then, you add equals 8 (=8)

So your equation looks like:

9x + 6 = 8

The p-value for the given mentioned test, with the test static of -2.01  is calculated out to be  0.0456.

To calculate the p-value for this test, we first need to determine the appropriate null and alternative hypotheses.

Null hypothesis: The proportion of customers who wait longer than 5 minutes at the drive-through for this franchise is equal to the national value of 12%.

Alternative hypothesis: The proportion of customers who wait longer than 5 minutes at the drive-through for this franchise differs from the national value of 12%.

We can use a two-tailed Z-test to test this hypothesis, with a significance level of alpha = 0.05.

The test statistic is given as -2.01. Since this is a two-tailed test, we need to find the area in both tails of the standard normal distribution that is at least as extreme as the test statistic.

Using a standard normal distribution table or a calculator, we find that the area to the left of -2.01 is 0.0228. The area to the right of 2.01 is also 0.0228. Therefore, the total p-value is the sum of these two probabilities:

p-value = 0.0228 + 0.0228 = 0.0456

Therefore, the p-value for this test is 0.0456. This means that there is a 4.56% chance of obtaining a test statistic as extreme as -2.01, assuming that the null hypothesis is true. Since the p-value is less than the significance level of 0.05, we reject the null hypothesis and conclude that the proportion of customers who wait longer than 5 minutes at the drive-through for this franchise differs from the national value of 12%.

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The value of XZ is approximately equal to 96.86.

What is inequality ?

An inequality is a mathematical statement that compares two values, expressions, or quantities using inequality symbols such as "<" (less than), ">" (greater than), "<=" (less than or equal to), ">=" (greater than or equal to), or "≠" (not equal to).

We can start by using the transitive property of equality to find that XY = YZ = 95 means that XY + YZ = XZ = 190.

Next, we can use the given information about WX and WZ to write an equation for XZ in terms of u. Since WZ = WX + XZ, we can substitute the given expressions to get:

7u = (u + 66) + XZ

Simplifying and solving for XZ, we have:

7u - u - 66 = XZ

6u - 66 = XZ

Now, we can substitute this expression for XZ into our earlier equation to get:

XZ = 190 = 95 + 95 = XY + YZ = (WX - 66) + (6u - 66)

Simplifying and solving for u, we get:

6u - 66 = 190 - WX

6u = 256 - WX

u = (256 - WX)/6

Substituting this value of u back into the expression for XZ, we get:

XZ = 6u - 66 = 6[(256 - WX)/6] - 66 = 190 - WX

Therefore, XZ = 190 - WX, where WX = u + 66 = (256 - WX)/6 + 66. We can solve for WX by multiplying both sides by 6:

6WX = 256 - WX + 396

7WX = 652

WX = 93.14 (rounded to two decimal places)

Substituting this value into our expression for XZ, we have:

XZ = 190 - WX = 190 - 93.14 = 96.86 (rounded to two decimal places)

Therefore, XZ is approximately equal to 96.86.

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Answer:)

O segment XW

Explanation:)

hope it helps :)

Brainliest pls:)

The correct inequality that can be used to determine g, the maximum number of gigabytes Rahul can use while staying within his budget is:

95 > 4g + 36

What is an inequality?

Rahul pays a flat cost of $36 per month and $4 per gigabyte. So, his monthly bill can be expressed as:

Bill = 4g + 36

where g is the number of gigabytes used in the month.

We want to find the maximum number of gigabytes Rahul can use while staying within his budget of $95 per month. This means we need to find the value of g that satisfies the inequality:

Bill ≤ 95

Substituting the expression for Bill, we get:

4g + 36 ≤ 95

Subtracting 36 from both sides, we get:

4g ≤ 59

Dividing both sides by 4, we get:

g ≤ 14.75

Since the number of gigabytes used cannot be a fraction, we can conclude that the maximum number of gigabytes Rahul can use while staying within his budget is 14. Therefore, the correct inequality is:

95 > 4g + 36

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The main difference between the reciprocal and inverse of a function is their definition and properties. The reciprocal of a function f(x) is defined as 1/f(x), while the inverse of a function f(x) is a function f^(-1)(x) such that [tex]f(f^(-1)(x))=x and f^(-1)(f(x))=x[/tex] for all x in their respective domains.

The domains of sine, cosine, and tangent functions need to be restricted to define their inverse functions because they are not one-to-one functions. In order to have an inverse, a function must be both one-to-one (each output has only one input) and onto (each output value is mapped by at least one input value).

For example:
- sine: restricted domain to [-π/2, π/2] to define its inverse, arcsin (sin^(-1))
- cosine: restricted domain to [0, π] to define its inverse, arccos (cos^(-1))
- tangent: restricted domain to (-π/2, π/2) to define its inverse, arctan (tan^(-1))

These restrictions ensure that the functions become one-to-one and onto, making it possible to define their inverses uniquely.

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The size of the x in the regular pentagon is 36 degrees.

In a regular pentagon, all the sides are congruent and all the angles are congruent. To find the internal angle of a regular pentagon, we can use the formula:

Interior angle = (n-2) x 180 / n

where n is the number of sides of the polygon.

For a regular pentagon, n = 5, so we can substitute this value into the formula:

Interior angle = (5-2) x 180 / 5

Interior angle = 3 x 180 / 5

Interior angle = 540 / 5

Interior angle = 108 degrees

For the value of x, we have

x = 108 degrees/3

Evaluate

x = 36 degrees

Therefore, the value of x in the regular pentagon is 36 degrees.

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x=-4 or -3 is simplified for x²+12/x²-16. This is because when the numerator and denominator are both set equal to zero, the only value of x that will satisfy both equations is x=-4 or -3.

The Zero Product Property states that if the product of two real numbers is equal to zero, then at least one of the two numbers must be zero.

When solving an equation like x²+12/x²-16 with x=-4, we must use the Zero Product Property, which states that if the product of two factors is equal to zero, then at least one of the two factors must be equal to zero.

In this case, we can set the numerator and denominator of the equation equal to zero and solve:

x²+12=0

x²-16=0

We can solve each of these equations separately by factoring:

x²+12=0

(x+3)(x+4)=0

x+4=0, x+3=0

x=-4 or -3

x²-16=0

(x-4)(x+4)=0

x-4=0

x=4

Therefore, x=-4 or -3 is the correct answer for x²+12/x²-16.

This is because when the numerator and denominator are both set equal to zero, the only value of x that will satisfy both equations is x=-4 or -3.

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When the side of the ice cube is 6 cm, it is decreasing at a rate of approximately -0.0139 cm/sec.

To find the rate at which a side of the cubic ice cube is decreasing when the side is 6 cm, we need to use the chain

rule from calculus.

Let V be the volume of the cube and s be the side length. We know that [tex]V = s^3.[/tex]

Differentiate both sides of the equation with respect to time (t). This gives us [tex]dV/dt = 3s^2 × ds/dt.[/tex]

Plug in the given values. We know that dV/dt = -1.5 cm³/sec (since the volume is decreasing) and s = 6 cm.

Solve for ds/dt, which represents the rate at which a side of the ice cube is decreasing.

[tex]-1.5 = 3(6^2) × ds/dt[/tex]

Now, divide both sides by [tex]3(6^2)[/tex]:

ds/dt = -1.5 / (3 × 36)

ds/dt = -1.5 / 108

ds/dt ≈ -0.0139 cm/sec

So, when the side of the ice cube is 6 cm, it is decreasing at a rate of approximately -0.0139 cm/sec.

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[tex] \: \: [/tex]

I believe the equation you meant to write is:

√(8x) = 6

To solve for x, we need to isolate x on one side of the equation. We can do this by squaring both sides:

(√(8x))^2 = 6^2

Simplifying the left side, we get:

8x = 36

Dividing both sides by 8, we get:

x = 4.5

Therefore, the solution to the equation √(8x) = 6 is x = 4.5.

Since m and k(k+1)/2 are both integers, we can conclude that (k+1)(k+2)(k+3) is an integer multiple of 6. Thus, by the principle of mathematical induction, the formula n(n+1)(n+2) is an integer multiple of 6 for all non-negative integers n

Mathematical induction is a proof technique used to establish a statement for all positive integers by proving it for the base case and showing that if the statement holds for an arbitrary integer, it must also hold for the next integer. It is a powerful method to prove mathematical statements that follow a pattern or recursive structure.

To prove by mathematical induction that n(n+1)(n+2) is an integer multiple of 6 for all non-negative integers n, we will first show that the formula holds true for the base case n=0.

Base case:

When n = 0, we have:

0(0+1)(0+2) = 0 * 1 * 2 = 0, which is an integer multiple of 6 since 0 = 6 * 0.

Induction hypothesis:

Assume that for some k >= 0, k(k+1)(k+2) is an integer multiple of 6.

Induction step:

We need to show that the formula holds true for k+1, assuming that it holds true for k. That is, we need to show that (k+1)(k+2)(k+3) is also an integer multiple of 6.

Expanding the formula, we get:

(k+1)(k+2)(k+3) = (k² + 3k + 2)(k+3) = k³ + 6k² + 11k + 6

Now, we can use the induction hypothesis that k(k+1)(k+2) is an integer multiple of 6 to write k(k+1)(k+2) = 6m, where m is some integer. Substituting this into the above equation, we get:

k³ + 6k² + 11k + 6 = 6m + k³ + 3k² + 2k

Factoring out 3k² + 3k, we get:

k³ + 6k² + 11k + 6 = 6m + 3k(k+1) + 2k

Factoring out 2k from the last two terms, we get:

k³ + 6k² + 11k + 6 = 6m + 3k(k+1) + 2k = 6(m + k(k+1)/2)

Since m and k(k+1)/2 are both integers, we can conclude that (k+1)(k+2)(k+3) is an integer multiple of 6. Thus, by the principle of mathematical induction, the formula n(n+1)(n+2) is an integer multiple of 6 for all non-negative integers n

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The two equations represent two lines in the plane that are parallel and never intersect, so there is no point that satisfies both equations simultaneously. No solution equations.

In mathematics, an equation is a statement that two expressions are equal. It consists of two sides separated by an equal sign (=). The expression on the left side of the equal sign is usually called the left-hand side (LHS), while the expression on the right side is called the right-hand side (RHS).

Equations can take many different forms and can involve various types of functions and operators, such as addition, subtraction, multiplication, division, exponentiation, logarithms, trigonometric functions, and more. They can also involve one or more variables, which can be solved for to obtain a specific value or range of values that make the equation true. Equations are used extensively in mathematics, science, engineering, economics, and many other fields.

We can solve this system of equations using the substitution method by solving one of the equations for one variable in terms of the other, and then substituting that expression into the other equation.

Let's solve the second equation for y in terms of x:

-x + y = 4

y = x + 4

Now we substitute this expression for y into the first equation and solve for x:

-2x + 2y = 7

-2x + 2(x + 4) = 7

-2x + 2x + 8 = 7

8 = 7

This last equation is a contradiction, which means that there is no solution to the system of equations. Geometrically, the two equations represent two lines in the plane that are parallel and never intersect, so there is no point that satisfies both equations simultaneously.

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