Question 4(Multiple Choice Worth 2 points)
(Factoring Algebraic Expressions LC)
Rewrite 8x + 64 using a common factor.
O8x(x + 64)
8x(x + 8)
O 8(x+64)
8(x + 8)

Algebraically, we can write the solution set of the system of inequalities as follows:   -2x - 3 ≤ y ≤ 3x 2  and  y = -5 satisfies this inequality, the point (1, -5) is in the solution set of the system of inequalities

An algebraic expression is an expression made up of constant algebraic numbers, variables and  algebraic operations (addition, subtraction, multiplication, division and exponentiation, which is a rational number).

To determine the relationship between the point (1, -5) and the given system of inequalities, we must substitute the values ​​of x and y in each inequality and check whether the point satisfies the inequality or not. 

 For the first inequality, we have: 

 y ≤ 3 x 2 

 Substituting x = 1 and y = -5, we get: 

 -5 ≤ 3 (1) 2 

 -5 ≤ 5 

 This inequality is true, so the point (1, -5) satisfies this inequality.  

 For the second inequality, we have: 

 y ≥ -2x -3 

 Substituting x = 1 and y = -5, we get: 

 -5 ≥ -2(1) -3 

 -5 ≥ -5 

 This inequality is also true, so the point (1, -5) also satisfies this inequality. 

 Since the point (1, -5) satisfies both inequalities, it lies in the region that satisfies the system of inequalities. Geometrically, the point (1, -5) lies in the shaded area between the two lines y = 3x 2 and y = -2x -3 in the xy plane. Algebraically, we can write the solution set of the system of inequalities as follows: 

 -2x - 3 ≤ y ≤ 3x 2 

 Substituting x = 1, we get: 

 -2 (1) - 3 ≤ y ≤ 3 (1) 2 

 -5 ≤ y ≤ 5 

 Since y = -5 satisfies this inequality, the point (1, -5) is in the solution set of the system of inequalities

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To determine the number of triangles in any given level of the pyramid, we can use the formula:

Number of triangles = (level)^2

Therefore, to find the number of triangles in the first level of the pyramid, we can substitute 1 for level in the formula:

ƒ(1) = (1)^2 = 1

So, there is only 1 triangle in the first level of the pyramid.

Answer:

1

Step-by-step explanation:

The speed of the train is 40 km/h. and the length and breadth of the plot are 33 meters and 16 meters respectively.

The solution is as follows :-

(i) Let's assume that the breadth of the rectangular plot is x meters. Then, according to the problem, the length of the plot is (2x + 1) meters. The area of the plot is given as 528 m². We know that the area of a rectangle is given by the product of its length and breadth. So we can write:

Area = length x breadth

528 = (2x + 1) x x

Simplifying this equation, we get:

528 = 2x² + x

2x² + x - 528 = 0

This is a quadratic equation in the variable x. We can solve this equation using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

where a = 2, b = 1, and c = -528. Plugging in these values, we get:

x = (-1 ± √(1² - 4(2)(-528))) / 2(2)

x = (-1 ± √(1 + 4224)) / 4

x = (-1 ± 65) / 4

The negative value of x does not make sense in the context of the problem. So we can discard it and take the positive value of x:

x = 16

This means that the breadth of the plot is 16 meters. The length of the plot is given as (2x + 1), which evaluates to 33 meters.

So the length and breadth of the plot are 33 meters and 16 meters respectively.

(ii) Let's assume that the speed of the train is x km/h. According to the problem, the distance traveled by the train is 480 km. We know that speed is defined as distance traveled per unit time. So the time taken by the train to cover the distance of 480 km at a speed of x km/h is given by:

time = distance / speed

time = 480 / x

If the speed of the train had been 8 km/h less, the time taken to cover the same distance would have been 3 hours more. So we can write another equation for the time taken in this case:

time + 3 = 480 / (x - 8)

Now we can set these two equations equal to each other, since they both represent the time taken to cover the same distance:

480 / x = 480 / (x - 8) + 3

Simplifying this equation, we get:

480(x - 8) = 480x + 3x(x - 8)

480x - 3840 = 480x + 3x² - 24x

3x² - 24x - 3840 = 0

Dividing both sides by 3, we get:

x² - 8x - 1280 = 0

This is a quadratic equation in the variable x. We can solve this equation using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

where a = 1, b = -8, and c = -1280. Plugging in these values, we get:

x = (8 ± √(64 + 5120)) / 2

x = (8 ± √5184) / 2

x = (8 ± 72) / 2

The negative value of x does not make sense in the context of the problem. So we can discard it and take the positive value of x:

x = 40

Therefore, the speed of the train is 40 km/h.

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The conditional probability that the coin chosen is fake given that it came up heads after flipping is not possible to determine without additional information.

The given information provides us with the result of a single trial, and it does not provide any information about the prior probability of choosing a fake coin from the bag. Therefore, the conditional probability of the coin being fake given that it came up heads depends on the prior probability of choosing a fake coin, which is not given.

If we assume that there is an equal chance of choosing a real or fake coin from the bag, then the probability that the coin is fake would be the ratio of the probability of choosing a fake coin to the probability of getting heads from either a real or a fake coin.

However, if we assume that the prior probability of choosing a fake coin is very low, then the probability of the coin being fake would also be very low, even if it came up heads. Therefore, without information about the prior probability of choosing a fake coin, we cannot determine the conditional probability that the coin is fake given that it came up heads.

Therefore, the given information is not sufficient to determine the conditional probability that the coin is fake

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The total amount spent by Tom in the cinema with his family is £33.50

The given question is based on the concept of arithmetic operations, where we have to find the total expenditure of Tom in the cinema with his family. According to the given information, he bought 3 child tickets at £5.50 each and 2 adult tickets for £8.50 each.

Therefore, the total cost spent by him would be calculated as follows:

Total cost of 3 child tickets = 3 × £5.50 = £16.50

Total cost of 2 adult tickets = 2 × £8.50 = £17

Total expenditure of Tom = £16.50 + £17 = £33.50

Therefore, the total amount spent by Tom in the cinema with his family is £33.50.

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The equation representing the total number of pages read by Victor is: p =  9 + 2m.

An indicator of how some quantity changes in proportion to another is called a rate of change.

Positive or negative change rates are possible. This reflects a change in the mathematical model -value between the two points of information. Zero change rate refers to a quantity that does not fluctuate over time.

You can determine the amount of change that has occurred over this time period if you have data at two points in time. The outcome is known as the rate of change, sometimes known as the percentage change, and is represented by a percentage (in absolute numbers, it really is merely a difference).

Given data:

Let 'm' be the total number of minute.

Let 'p' be the total number of pages read after 'm' minutes.

p = 9 + 2m

Thus, the equation representing the total number of pages read by Victor is: p =  9 + 2m.

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The coordinates of the image point A" after the given series of transformations are (6,9,2), which corresponds to answer choice (b) (8,6).

A series of transformations is a sequence of two or more geometric transformations applied one after the other to a given figure or object, to produce a new transformed figure or object.

Each transformation in the sequence modifies the position, shape, or orientation of the figure or object in some way.

To find the image of point A(9,-5,2) after the given series of transformations, we can apply them one by one.

Reflection across the x-axis:

1. The reflection of point across the x-axis is obtained by changing the sign of its y-coordinate while keeping the x and z coordinates the same. Therefore, the image of A after reflection across the x-axis is A'(9,5,2).

2. Translation by the rule (x-3, y+4):

This transformation moves the point A' 3 units to the left and 4 units up. Therefore, the coordinates of the image point A" are obtained by subtracting 3 from the x-coordinate and adding 4 to the y-coordinate of A', i.e.,

A" = (9-3, 5+4, 2) = (6,9,2)

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The required probability for the given rate of calls per hour using Poisson distribution is ,

Probability of calls rate are 20 per hour is 0.734

Probability of calls rate are 30 per hour is 0.654..

Probability of 20 per hour is higher in comparison of 30 per hour.

Yes . Manager should consider second operator.

Actual rate of calls =  20 per hour,

Expected number of calls in a 15-minute period is,

= (20/60) × 15

= 5

The probability of getting 9 or more calls in a 15-minute period is,

P(X ≥ 9) = 1 - P(X ≤ 8)

Using the Poisson distribution with λ = 5, we have,

P(X ≤ 8)

= e^(-5) × (5^0/0!) + e^(-5) ×(5^1/1!) + ... + e^(-5) × (5^8/8!)

= 0.266

Probability of getting 9 or more calls in a 15-minute period is,

P(X ≥ 9)

= 1 - P(X ≤ 8)

= 1 - 0.266

= 0.734

If the rate of calls is really 30 per hour.

Expected number of calls in a 15-minute period is,

= (30/60) × 15

= 7.5

Probability of getting 9 or more calls in a 15-minute period is,

P(X ≥ 9) = 1 - P(X ≤ 8)

Using the Poisson distribution with λ = 7.5, we have,

P(X ≤ 8)

= e^(-7.5) * (7.5^0/0!) + e^(-7.5) * (7.5^1/1!) + ... + e^(-7.5) * (7.5^8/8!)

= 0.346

Probability of getting 9 or more calls in a 15-minute period is,

P(X ≥ 9)

= 1 - P(X ≤ 8)

= 1 - 0.346

= 0.654

Based on the calculations in parts a and b,

Probability of getting 9 or more calls in a 15-minute period is higher .

If the rate of incoming calls is 20 per hour (0.734) compared to if the rate is 30 per hour (0.654).

Based on the calculations,

Current operator can handle the call volume with a high probability if the call rate is 20 per hour.

However, if the call rate is 30 per hour, there is a relatively high probability of needing a second operator.

Manager should consider hiring a second operator if the rate of incoming calls is expected to be 30 per hour or higher.

Therefore, the probability of the given situation are,

For the rate of calls 20 per hour is 0.734.

For the rate of calls 30 per hour is  0.654.

Probability of 20 per hour is higher than 30 per hour.

Yes , manager should consider second operator is call rate are 30 per hour or higher.

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Answer:

34.54 in

Step-by-step explanation:

The approximate circumference of a plate with a diameter of 11 inches can be calculated using the formula C = πd, where C is the circumference and d is the diameter .

So, substituting the given value of diameter d = 11 inches and taking π to be approximately 3.14, we get:

C = πd = 3.14 x 11 = 34.54 inches (approx.)

Therefore, the approximate circumference of the plate is 34.54 inches.

Since it is required to round the answer to the nearest hundredth, we get the rounded answer as 34.54 rounded off to two decimal places, which is 34.54 (no rounding needed).

Answer: For circumference, you simply multiply the diameter by pi, so the answer you are looking for is 34.54

Step-by-step explanation:

Step-by-step explanation:

If you mean what are the solutions (zeroes or 'roots') :

the graph is equal to zero at x = 2 and 5  

  this is where the graph crosses the x-axis

the quadratic would be   f(x) = (x-2)(x-5)  = x^2 -7x+10

The interest rate of Nick's investment is 3% per year.

We can use the formula for compound interest to solve for the interest rate r:

A = P(1 + r/n)^(nt)

r = n[ (A/P)^(1/nt) - 1 ]

Given that:

A = final amount = $97,153.41

P = principal amount = $74,460

n = number of times compounded annually = 1 (compounded annually)

t = time in years = 9

r = annual interest rate r = ?

Solving for rate r as a decimal

r = n[ (A/P)^(1/nt) - 1 ]

r = 1 × [ (97153.41/74640)^(1/1×9) - 1]

r = 0.0297237

Then convert r to R as a percentage

R = r × 100%

R = 0.0297237 × 100%

R = 3% per year.

Therefore, the interest rate is 3%.

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To find the probability that a student break lasts more than 7 minutes, we can use the z-score formula to standardize the break time and then use the standard normal distribution table or a calculator to find the probability. The probability that a student break lasts more than 7 minutes is approximately 0.3745 or 37.45%.

Step 1: Calculate the z-score.
The z-score formula is given by:
z = (X - μ) / σ
where X is the break time, μ is the mean, and σ is the standard deviation. In this case, X = 7 minutes, μ = 6.3 minutes,

and σ = 2.2 minutes. Plugging in these values:
z = (7 - 6.3) / 2.2 ≈ 0.32
Step 2: Find the probability using the standard normal distribution table or a calculator.
Now that we have the z-score, we can find the probability that a student break lasts more than 7 minutes. We want to find the area to the right of the z-score (0.32) since we are interested in break times longer than 7 minutes.
Using a standard normal distribution table or a calculator, we find that the area to the left of z = 0.32 is approximately 0.6255. However, we want the area to the right of z = 0.32, which is given by:
Area to the right of z = 1 - Area to the left of z
Area to the right of z ≈ 1 - 0.6255 ≈ 0.3745.

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The probability that the card that melody turns over has a rose on it = 2/3

Here, 3 cards with a picture of a rose and 1 card with a picture of a daisy.

So, the total number of cards : 3 + 1 = 4

First Melody  turns over one card face up and finds the picture of a rose on it.

Now there are 3 cards( two rose cards and a card with a picture of a daisy) on a table.

Let us assume that event A:  the card that melody turns over (second time) has a rose on it

here, the number of rose cards n(A) = 2

And the number of remaining cards n(S) = 3

Using the definition of probability, the probability of this event would be,

P(A) = n(A)/n(S)

P(A) = 2/3

Therefore, the required probability is 2/3

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The November weight is higher than the original mean weight, the new mean weight has increased. It increased by 0.21 lbs. So, correct option is A.

To determine how the November weight of 13.9 lbs affects the mean weight of the cat, we need to calculate the new mean weight with the updated data.

To do this, we first need to find the sum of all the weights, including the November weight:

Sum of weights = 13.5 + 12.5 + 10.5 + 11.2 + 11.2 + 10.8 + 9.5 + 13.4 + 11.5 + 11.8 + 13.9 = 129.8

Next, we need to find the new number of data points, which is 11 (the original 10 months plus the additional November weight).

Now, we can calculate the new mean weight by dividing the sum of weights by the number of data points:

New mean weight = Sum of weights / Number of data points = 129.3 / 11 = 11.8 lbs

The original mean weight was 11.59 lbs. So, the difference in mean weight after adding the November weight is:

Difference = New mean weight - Original mean weight = 11.8 - 11.59 = 0.21 lbs

Therefore, the correct answer is A.

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Complete question is:

The weight of a cat for 10 months is shown in the table.  In November, the cat weighed 13.9 lbs.  How did this affect the mean weight of the cat?

Month:       January February March April May June July August September October

Weight (lbs): 13.5 12.5 10.5 11.2 11.2 10.8 9.5 13.4 11.5 11.8

A. It increased by 0.21 lbs

B. It decreased by 0.21 lbs

C. It increased by 1.38 lbs

D. It decreased by 1.38 lbs

The probability that the random variable will assume a value between 45 and 55 is given by: 0.6827

A random variable is normally distributed.

Mean μ = 50

Standard deviation σ = 5

According to the empirical rule, also known as 68-95-99.7 rule, the percentage of values that lie within an interval with 68%, 95% and 99.7% of the values lies within one, two or three standard deviations of the mean of the distribution.

The more precise statement for 68 percent is:

[tex]P(\mu-\sigma < X < \mu+\sigma) = 68.27%[/tex]

Since the interval of interest given in question is (45,55), it can be rewritten as  (50-5, 50 + 5)

Thus we have:

P(50 - 5 < X < 50+5) = 68.27%

P(45 < X < 55) = 0.6827.

Thus, 0.6827 is the probability that the random variable will assume a value between 45 and 55.

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The probability that the risk level is 0 irrespective of the applicant's marital status is 0.6.

The given information states that 60% of loan applicants were classified as low risk (risk level 0), and 40% were classified as high risk (risk level 1). Therefore, the probability of an applicant being classified as low risk is 0.6, and the probability of being classified as high risk is 0.4.

The question asks for the probability that an applicant is classified as low risk regardless of their marital status. This means that the probability is not affected by whether the applicant is married or not.

Since we know that the probability of an applicant being classified as low risk is 0.6, and this probability is not affected by their marital status, the answer to the question is simply 0.6.

In other words, regardless of whether an applicant is married or not, there is a 60% chance that they will be classified as low risk.

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Complete question is:

In a study, 60% of loan applicants were classified as low risk (risk level 0), while 40% were classified as high risk (risk level 1). And 40%of loan applicants were classified as low risk and married and 30% of loan applicants were classified as high risk and married. What is the probability that an applicant is classified as low risk (risk level 0) regardless of their marital status?

Answer:

To combine like terms, we add the coefficients of the variable n. Therefore, we can combine the two terms with n: 4n + 2n = 6n The expression becomes: 6n + 11

The output of the function g(x) when x = 4 is 9.

If you input 4 into g(x), we get:

g(4) =  = [tex]3^{4/2}[/tex] = 9

An explicit function in mathematics is one that may be explicitly computed for any given value of its independent variable, typically written as x. In other words, an explicit function does not include any additional variables or unknowns and simply expresses the dependant variable (y) as the independent variable (x). Often, it is expressed as an algebraic formula or equation that can be evaluated for any given value of x. Calculus, differential equations, and statistics are just a few of the mathematics disciplines where explicit functions are helpful. They make it simpler to analyse and resolve mathematical issues and offer a clear approach to define a relationship between two variables. In programming, explicit functions are also employed.

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No Solutions

5−4+7x+1 =

x + 2

One Solution

5−4+7x+1 =

x - 2

Answer:=-3

Step-by-step explanation: trust the processe

-3x-3<6

   +3 l+3

  -3x  > 9

  -3x  -3x

  x=-3

In the case of the property "if c² =0, then c=0", the set Z and Q do not fulfill this property, while the set 4Z does.

To determine if a set of numbers fulfills a property, you need to evaluate whether every element in the set satisfies that property. In the case of the property "if c² =0, then c=0", this means that for any element c in the set, if c² =0, then c must be equal to 0.

The set of integers does not fulfill this property because there are elements in Z that satisfy c² =0 without being equal to 0. For example, 2 × 0=0, but 2 is not equal to 0.

Similarly, the set Q (the set of rational numbers) also does not fulfill this property because there are rational numbers that satisfy c² =0 without being equal to 0. For example, (1/2) × (1/2)=1/4, which is not equal to 0.

However, the set 4Z the set of integers that are multiples of 4 does fulfill this property because every element in 4Z can be written as 4n for some integer n, and if (4n)² =0, then 4n=0, which implies that n=0, and hence c=4n=0.

and in the case of the property "if c² =0, then c=0", the set Z and Q do not fulfill this property, while the set 4Z does.

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To be 95% confident that the true proportion of U.S. voters who oppose capital punishment is estimated to within 2%, the pollster should survey 2401 voters.

The pollster wants to estimate a proportion, so we will use the formula for sample size for proportions:

[tex]n=\frac{z^2 * p * (1 - p)}{(E^2}[/tex])

where:

n = sample size

z = the z-score associated with the desired confidence level (95% confidence level corresponds to z = 1.96)

p = an estimate of the true proportion (we don't know this yet, so we will use 0.5 as a conservative estimate that gives the largest sample size)

E = the desired margin of error (2% = 0.02)

Substituting the values into the formula, we get:

[tex]n = \frac{(1.96^2 * 0.5 * (1 - 0.5))} {(0.02^2} = 2401[/tex]

Therefore, the pollster should survey 2401 voters to be 95% confident that the true proportion of U.S. voters who oppose capital punishment is estimated to within 2%.

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Let's first draw the rhombus and label the diagonals:

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

     A

     o

    / \

3.5   12

/       \

o----x----o

\       /

3.5   12

  \   /

   o B

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

Let the rhombus be ABCD, with AB = BC = CD = DA. Let O be the center of the circle, which is also a vertex of the rhombus. Then, OA and OB are radii of the circle, and they are also perpendicular bisectors of sides AB and BC, respectively. Therefore, triangle AOB is a right triangle, and we can use the Pythagorean Theorem to find the length of OB:

OA = OB = OC = OD (since O is the center of the circle)

AB = BC = 12 (since 12 is the length of diagonal AC)

AO^2 = AB^2/4 + OB^2 (since AO and OB are the legs of right triangle AOB)

Substituting AB = 12 and simplifying, we get:

OB^2 = AO^2 - AB^2/4

= (3.5/2)^2 - 12^2/4

= 49/16 - 144/4

= 49/16 - 36

= 1/16

Taking the square root of both sides, we get:

OB = \sqrt{1/16} = 1/4

Now, the circle is tangent to sides AB and BC, so its diameter must be perpendicular to these sides. Therefore, the diameter of the circle is equal to the length of diagonal BD, which is the hypotenuse of right triangle AOB:

BD^2 = AB^2 + OB^2

= 12^2 + (1/4)^2

= 144 + 1/16

= 577/16

Taking the square root of both sides, we get:

BD = \sqrt{577}/4

Finally, the area of the circle is given by:

A = pi*(BD/2)^2

= pi*(\sqrt{577}/8)^2

= pi*577/64

Therefore, the exact value of the circle area is (577/64)*pi.

In the context of our theory of inductive proofs, P(n) represents the quantity about which we are proving something- False.

A proof by induction requires justification at every step, just like a regular proof. But, it uses a clever technique that enables you to demonstrate the truth of a proposition when n is 1, assume it is true for n=k, and then demonstrate that it is true for n=k+1.

According to the theory, all that is required to demonstrate a person's ability to ascend to the nth floor of a fire escape is for them to demonstrate their ability to ascend the fire escape ladder (n=1) and their ability to ascend the stairs from any level of the fire escape (n=k) to the next level (n=k+1).

You could have been asked to assume the n-1 case and prove the n case if you've done proof by induction before, or to assume the n case and show the n+1 case. This is the same as what I'm describing here, but I'll use a different letter since I believe it helps people understand what each instance is for better.

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Answer: Dilation

Step-by-step explanation: The Square in the bottom of the original shape has been made smaller, indicating that the shape has been dilated.

Answer:

m<1 = 60

m<2 = 60

m<3 = 30

Step-by-step explanation:

Since Beth is moving faster than Amy and has a steeper path than Amy, the given graph problem's answer indicates that Beth is moving faster than Amy.

Theoretical physicists use graphs to analyse and illustrate assertions rather than values. A graph point typically depicts the connection between several different items. A specific type of transport system made up of groups or lines is called a graph.

Glue should be used to secure the channels or edges. Within the confines of this network were the digits [tex]1[/tex] through[tex]4[/tex] as well as the individuals[tex]2.5[/tex], some, or[tex]4.5[/tex]

a. To graph the information pertaining to Beth's journey, the duration can be plotted on the [tex]x- axis[/tex] and the distance can be plotted on the [tex]y- axis[/tex] The chart contains the following details:

[tex]time and distance = (5,10) (10,22)[/tex]

[tex]Slope = (22-10)/(10-5)[/tex] = [tex]12/5[/tex]

b.Beth moves at a speed of of [tex]\frac{12}{5}[/tex] [tex]feet/sec[/tex]

c. Amy moves at [tex]75 feet/sec[/tex]

[tex]rate = \frac{distance }{time}[/tex]

= [tex]\frac{75}{30}[/tex] = [tex]\frac{5}{2}[/tex][tex]feet/sec[/tex]

Amy's line has a slope of [tex]\frac{5}{2}[/tex] while Berth line has a slope [tex]\frac{12}{5}[/tex]  which is steeper

This shows that Beth is moving faster than Amy and that her route is more difficult.

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The relationship between velocity and time can be expressed as V = 5 + 0.5t  and the time taken is 20 seconds.

The velocity of a car is expressed as the sum of a constant part and a part that varies with time, and since the car has a constant acceleration, this varying part can be expressed as the product of acceleration and time.

I) Let Vc be the constant part of the velocity and Vv be the part that varies with time. Then we can express the velocity of the car as:

V = Vc + Vv

Since the car is moving with a constant acceleration, the varying part of the velocity can be expressed as:

Vv = at

Therefore, we can rewrite the velocity equation as: V = Vc + at

To find the relationship between the velocity and time taken, we can use the given values for V and t. Substituting t = 8s and V = 9 m/s, we get:

9 = Vc + 8a

Substituting t = 12s and V = 11 m/s, we get:

11 = Vc + 12a

We can solve these equations simultaneously to obtain the values of Vc and a. Subtracting the first equation from the second, we get:

2 = 4a

a = 0.5 m/s²

Substituting this value of an into the first equation, we get:

9 = Vc + 4

Vc = 5 m/s

Therefore, the relationship between the velocity and time taken is:

V = 5 + 0.5t

II) To find the time taken when the velocity is 15 m/s, we can use the velocity equation:

V = 5 + 0.5t

Substituting V = 15 m/s, we get:

15 = 5 + 0.5t

t = 20 seconds

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The statement " the vector spaces (consisting of all polynomial os degree at most 3) and are isomorphic" is true because linear transformation T from the vector space of polynomials of degree at most 3 with real coefficients to R^4 is an isomorphism.

It is true that the vector spaces consisting of all polynomials of degree at most 3 are isomorphic.

To see why, consider the linear transformation T from the vector space of polynomials of degree at most 3 with real coefficients to R^4 given by

T(a + bx + cx^2 + dx^3) = (a, b, c, d).

It can be shown that T is a linear transformation and an isomorphism. This means that T is a bijective linear transformation, which preserves the structure of the vector space.

Thus, the vector space of polynomials of degree at most 3 with real coefficients and R^4 are isomorphic, and therefore the original vector spaces consisting of all polynomials of degree at most 3 are also isomorphic.

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