Subtract. Simplify, if possible.
4 1/2-2 3/2
(34
Give your answer as a mixed number.

The amount of money the manufacturer should expect to pay in refunds is 32.76X dollars.

Let Y denote the life in years of a printer, which is exponentially distributed with mean 3 years. Therefore, the rate parameter λ of the exponential distribution is λ = 1/3, which is obtained from the formula of the exponential distribution, E(Y) = 1/λ = 3 years. The probability that a printer fails during the first year following its purchase is:

P(Y < 1) = F(1) = 1 - e-λt = 1 - e-1/3(1) = 0.2835

The probability that a printer fails during the second year following its purchase is:

P(1 < Y < 2) = F(2) - F(1) = e-1/3(2) - e-1/3(1) = 0.3219 - 0.2835 = 0.0384

The probability that a printer does not fail during the first two years is:

P(Y > 2) = 1 - F(2) = 1 - e-1/3(2) = 0.6797

Let X denote the refund payment in dollars that the manufacturer should pay to the buyer of a printer if the printer fails during the first year, and X/2 denote the refund payment if the printer fails during the second year. Then, the total refund payment per printer is R = XI(Y < 1) + XI(1 < Y < 2)/2 = XI(Y < 1) + (X/2)I(1 < Y < 2), where I(.) is the indicator function that takes the value of 1 if the condition in the parentheses is true and 0 otherwise. The expected value of the total refund payment per printer is:

E(R) = E[XI(Y < 1)] + E[(X/2)I(1 < Y < 2)]

= X(0.2835) + (X/2)(0.0384) = 0.3276X

Hence, the expected total refund payment for 100 printers is:

Expected total refund payment = 100E(R) = 100(0.3276X) = 32.76X dollars. The amount of money the manufacturer should expect to pay in refunds is 32.76X dollars.

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Based on the following calculator output, the mean of the dataset, rounding to the nearest 100th is 167.14

Mean value is defined as sum of all data values divided by the total number of data values,

using the given data output, we get:

Mean = x = ∑x/n

x = 1170/7

since its given to us that ∑x = 1170; n = 7

therefore, x = 167.14

(Rounded to 2 decimals)

Based on the following calculator output, the mean of the dataset, rounding to the nearest 100th is 167.14

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Answer:

Elise's average pay per hour for the year 2017 was £11.88.

Step-by-step explanation:

First, we need to calculate Elise's monthly salary in 2017 after the 3% increase:

3% of £2100 = 0.03 x £2100 = £63

Elise's new monthly salary in 2017 = £2100 + £63 = £2163

Next, we need to calculate her total earnings in 2017:

Total earnings = monthly salary x number of months worked

Elise worked for 45 weeks in 2017, which is approximately 10.4 months:

Total earnings = £2163 x 10.4 = £22,477.20

Finally, we can calculate her average pay per hour for the year:

Total number of hours worked = 42 hours/week x 45 weeks = 1890 hours

Average pay per hour = total earnings / total number of hours worked

Average pay per hour = £22,477.20 / 1890 hours = £11.88 per hour

Answer:

I got $13.73 per hour

Step-by-step explanation:

If you multiply 2100 by 12 months you get $25200 for an annual salary before percent increase.

Then you include the 3% salary increase each month 25200 x .03 = 756 then add the two 25200 + 756 = 25956 OR you can calculate the longer way by doing each individual month, either way it will be the same answer. 2100 x .03 = 63 then 2100 + 63 = 2163 then 2163 x 12 = 25956

Then you divide 25956 by the number of weeks worked in the year. 25956/45 = 576.8

So, each week $576.8 was earned.

Then to get an hourly rate, you divide this by the number of hours worked in each week. 576.8/42 = 13.73

So, the hourly rate is $13.73.

I hope this helped :)

Answer:

The island of Martinique has received $32,000 for hurricane relief efforts. The island’s goal is to fundraise at least y dollars for aid by the end of the month. They receive donations of $4500 each day. Write an inequality that represents this

situation, where x is the number of days.

Step-by-step explanation:

The total amount raised after x days is given by:

Total amount raised = $32,000 + $4,500x

We want this to be at least y by the end of the month. Assuming that there are 30 days in the month, we can write the inequality:

$32,000 + $4,500x ≥ y

Alternatively, if we don't want to assume the number of days in the month, we can use a variable for the number of days:

$32,000 + $4,500x ≥ y

This inequality states that the sum of the initial donation and the donations received each day multiplied by the number of days must be greater than or equal to the fundraising goal y.

The equilibrium concentration of HI(g) is 0.031 mole. It is less than the initial concentration of HI(g) which is 0.75 mole. Thus, the correct answer is less than.

At 1,000 K, the value of Ke for the reaction is 2.6 x 10-2, in an experiment, 0.75 mole of HI(g), 0.10 mole of of H2(g), H2(g), and 0.50 mole of 12(g) are placed in a 1.0 L container and allowed to reach equilibrium at 1,000 K. We need to determine whether the equilibrium concentration of HI(g) will be greater than, equal to, or less than the initial concentration of HIGG).

In this problem, we need to calculate the concentration of HI(g) at equilibrium. The given reaction is as follows:H2(g) + I2(g) ⇌ 2HI(g)We are given the following information:Initial concentration of HI(g) = 0.75 moleInitial concentration of H2(g) = 0.10 moleInitial concentration of I2(g) = 0.50 moleVolume of container = 1.0 LAt equilibrium, let's consider the concentration of HI(g) as x mole. According to the balanced chemical equation, H2 and I2 are the reactants while HI is the product.

Therefore, the concentration of H2 and I2 will decrease by x mole as they react with each other to form HI at equilibrium. Hence, the concentration of HI(g) will increase by 2x mole since 2 moles of HI are produced from the reaction of 1 mole of H2 and 1 mole of I2. Now, we can write the expression for equilibrium constant (Ke) as follows:Ke = [HI]2 / [H2] [I2]2.6 x 10-2 = (2x)2 / [(0.10 - x) (0.50 - x)]2.6 x 10-2 (0.10 - x) (0.50 - x) = 4x2(0.10 - x) (0.50 - x) = 4x2 - 2.6 x 10-2 (0.10 - x) (0.50 - x)8.54 x 10-3 x2 - 0.365 x + 0.02425 = 0x = 0.031 mole

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Runners and sprinters could be on the track team is 25 distance.

Distance:

Distance is a qualitative measurement of the distance between objects or points. In physics or common usage, distance can refer to a physical length or an estimate based on other criteria (such as "more than two counties"). The term Distance is also often used metaphorically to refer to a measure of the amount of difference between two similar objects.

According too the Question:

Based on the based Information:

15× 5÷3

canceling all the common factor, we get:

5 × 5 = 25

Now, the Product or Quotient is 25.

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The answer for question 63 is [tex]\frac{36}{7}[/tex] by using BODMAS rule and the answer for question 33 by simplifying is [tex]4a-27b[/tex].

What is BODMAS ?

BODMAS stands for "Bracket, Order, Division, Multiplication, Addition, and Subtraction," which is the order of operations used in mathematics to determine the sequence in which calculations are performed.

Question 63 [tex][(6+3)\div(-5-2)](-4)[/tex]

Now put  [tex]a=-5[/tex] and [tex]b=6[/tex]

⇒ [tex][(6+3)\div(-5-2)](-4)[/tex]

⇒ [tex][(9)\div(-7)](-4)[/tex]

⇒ [tex][\frac{9}{-7}] (-4)[/tex]

⇒ [tex]\frac{36}{7}[/tex]

Hence, the answer for question 63 is [tex]\frac{36}{7}[/tex] by using BODMAS rule.

Question 33  [tex](-5)*(3b-2a)-6(a+2b)[/tex]

⇒ [tex](-15b+10a)-6a-12b[/tex]

⇒ [tex]4a - 27b[/tex]

Hence, the answer for question 33 by simplifying is [tex]4a-27b[/tex].

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True, the x-intercept and the y-intercept will never be the same ordered pair because they have different x- and y-coordinates

Explanation:The statement is true. The x-intercept and the y-intercept for a line will never be the same ordered pair because they are two distinct points. The x-intercept is the point where a line crosses the x-axis, which means that the y-coordinate of this point is zero.

The y-intercept, on the other hand, is the point where the line crosses the y-axis, which means that the x-coordinate of this point is zero. Therefore, the x-intercept and the y-intercept will never be the same ordered pair because they have different x- and y-coordinates.

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The division of 4x³ + 14x² + 13x + 11 by x + 2 will have a quotient of 4x² + 14x + 1 and a remainder of 8 using synthetic division and can be expressed as (4x² + 14x + 1) + 9/(x + 2).

The procedure for synthetic division involves the following steps:

Divide.

Multiply.

Subtract.

Bring down the next term, and

Repeat the process to get zero or arrive at a remainder.

We shall divide 4x³ + 14x² + 13x + 11 by x + 2 as follows;

4x³ divided by x equals 4x²

x + 2 multiplied by 4x² equals 4x³ + 8x²

subtract 4x³ + 8x² from 4x³ + 14x² + 13x + 11 will give us 6x² + 13x + 11

6x² divided by x equals 6x

x + 2 multiplied by 6x equals 6x² + 12x

subtract 6x² + 12x from 6x² + 13x + 11 will give us x + 11

x divided by x equals 1

x + 2 multiplied by 1 equals x + 2

subtract x + 2 from x + 11 will give result to a remainder of 9.

Therefore by synthetic division, 4x³ + 14x² + 13x + 11 by x + 2 will result to a quotient of 4x² + 14x + 1 and a remainder of 9.

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Step-by-step explanation:

9. f(x) is x² so -4² will give you 16 under the f(x)

-4= 16

-2= 4

0= 0

1= 1

6 = 36

10. f(x) is x² -4 therefore f(x) is (-3)² -4= 5

-3= 5

-1= -3

0= -4

2= 0

5= 21

The expected value (μ) of the probability distribution of the discrete random variable X is μ = 7.26.

The correct option is B.

In probability theory, the expected value (or mean) is a measure of the central tendency of a random variable. It represents the average value that we would expect to observe if we repeated an experiment or random process many times.

Now, The expected value (μ) is calculated as:

μ = Σ(x P(X = x))

Let's calculate the expected value using the given probability distribution:

= (1 x 0.23) + (3 x 0.09) + (5 x 0.05) + (7 x 0.01) + (9 x 0.30) + (11 x 0.21) + (13 x 0.11)

= 0.23 + 0.27 + 0.25 + 0.07 + 2.7 + 2.31 + 1.43

= 7.26

Therefore, the expected value (μ) of the probability distribution is μ = 7.26.

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Answer:

We can use the Pythagorean theorem to find the length of side R:

R^2 = 15^2 + 14^2

R^2 = 225 + 196

R^2 = 421

R = √421

So, we have:

sin S = S/15

cos R = 14/√421

sin S/cos R = (S/15)/(14/√421) = S/(15*14/√421) = S/(210/√421) = S√421/210

Therefore, the expressions for sin S and cos R in simplest terms are:

sin S = S/15

cos R = 14/√421

sin S/cos R = S√421/210

The probability that the ticket picked up is a multiple of 3 or 5 is 0.47 or 47%.

To find the probability that the ticket picked up is a multiple of 3 or 5, we need to count the number of tickets that are multiples of 3 or 5, and then divide by the total number of tickets (100).

Multiples of 3: There are 33 multiples of 3 between 1 and 100 (3, 6, 9, ..., 99).

Multiples of 5: There are 20 multiples of 5 between 1 and 100 (5, 10, 15, ..., 100).

However, we have counted the multiples of 15 twice (since they are multiples of both 3 and 5), so we need to subtract them once to avoid double counting. There are 6 multiples of 15 between 1 and 100 (15, 30, 45, ..., 90).

Therefore, the total number of tickets that are multiples of 3 or 5 is

33 + 20 - 6 ⇒ 47.

The probability that the ticket picked up is a multiple of 3 or 5 is therefore:

P(multiple of 3 or 5) = number of multiples of 3 or 5 / total number of tickets

P(multiple of 3 or 5) = 47 / 100

P(multiple of 3 or 5) = 0.47

So the probability that the ticket picked up is a multiple of 3 or 5 is 0.47 or 47%.

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Answer:

Step-by-step explanation:

what?

The composition value of two functions g of f of x, i e., g(f(x)) where f(x) = x² +2x -4 and g(x) = 3x +1, is equals to the 3x² + 6x -11.

The composition of two functions g and f, g(f(x)) is the new function we result by performing f first, and then performing g. It can be written as (g º f)(x). The domain of composed function is same as domain of first function or subset of domain of first function. Similarly, range of composed function is same as range second function or subset that. The g(f(x)) can be solved by first we apply f, then apply g to that result. We have f(x) = x² + 2x -4 and g(x) = 3x + 1 and we have to determine g(f(x)). Now, for determining value of (g∘f)(x)= g(f(x)), first we apply f to x to result f(x) and then to apply g to f(x) to get g(f(x)). That is f to x= f(x) = x² + 2x - 4

and g to f(x), we replace x in g(x) by f(x) for g(f(x)).

=> g(f(x)) = 3 f(x) + 1

=> g(f(x)) = 3 ( x² + 2x -4) + 1

= 3x² + 6x - 12 + 1

= 3x² + 6x -11

Hence, required value is 3x² + 6x -11.

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The value of x -8.

An equation is a mathematical statement containing two algebraic expressions flanked by equal signs (=) on either side.

It shows that the relationship between the left and right printed expressions is equal.

All formulas hav LHS = RHS (left side = right side).

You can solve equations to determine the values ​​of unknown variables that represent unknown quantities.

If a statement does not have an equals sign, it is not an equation. A mathematical statement called an equation contains the symbol "equal to" between two expressions of equal value.

Hence, the value of x -8.

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Problem 1 (3 pts). The nearest neighbor method is used with the following labeled training data in a two class, two feature problem. Show clearly the decision boundaries obtained. Indicate all break points and slopes correctly Class1={(0,0)^T,(0,1)^T}. Class2={(1,0)^T,(0,0.5)^T}.

The decision boundary is the line that separates the two classes. When it comes to the nearest neighbour method, the boundary of a class is determined by the closest data point in the other class.What is the nearest neighbour method?The nearest neighbour method (NNM) is a type of lazy learning method in which the data is stored and the computation is deferred until the classification phase.

The goal of the nearest neighbor technique is to use the pattern of the closest data point to classify a new sample. It's also one of the simplest non-parametric classification methods, and it's based on the assumption that the feature spaces used by each class are continuous regions.For Class 1, there are two labeled training data points: (0, 0)T and (0, 1)T. Similarly, for Class 2, there are two labeled training data points: (1, 0)T and (0, 0.5)T.

To generate decision boundaries, follow the steps below:Step 1: Plot the given labeled training data. They are shown in the figure below. Step 2: Label the data points according to their class: Class 1 and Class 2.Step 3: Now, we need to find the nearest neighbors. Find the nearest neighbor for each of the labeled training data points from the opposite class. The distances are calculated as follows:For the Class 1 data points, the nearest neighbor is Class 2's (1, 0)T data point.

For the Class 2 data points, the nearest neighbour is Class 1's (0, 1)T data point.Step 4: Connect the nearest neighbour's labeled training data points with a line. These are the decision boundaries. The red line represents the decision boundary between Class 1 and Class 2. The green line represents the decision boundary between Class 2 and Class 1. The slopes of the decision boundaries are -1 for the red line and 1 for the green line. These slopes are the result of the negative reciprocal of the nearest neighbour's slope. The break point for the red line is 0.5, while the break point for the green line is 0. A break point is the point at which the decision boundary intersects the y-axis.

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Answer:

13

Step-by-step explanation:

distance formula:

=[tex]\sqrt{(y2-y1)^{2}+(x2-x1)^{2} }[/tex]

=[tex]\sqrt{(5--2)^{2}+(-5-6)^{2} } \\\sqrt{170} \\13[/tex]

The compound interest for the principal value $400, interest rate 7% and compounded anually for 3 years is equals to the $90.2

Compound interest is defined as the interest earn on interest, i.e., here interest on interest . It is denoted by CI and calculated by the below formula, CI

= Amount - principal

and A = P( 1 + r/n)ⁿᵗ

where P --> principal

A --> amount

r --> interest rate

n --> the number of times interest is compounded per year

t --> time in years

Now, we have principal, P = $400

interest rate, r = 7%

time, t = 3 years

here, compounded anually for 3 years so, n= 1

Substitute all known values in above formula,

A = P( 1 + r/n)ⁿᵗ

=> A = 400( 1 + 7/100)³

=> A = 400( 107/100)³

=> A = $490.02.

So, compound interest = A - P

=> CI = $90.2

Hence, required value is $90.2.

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Complete question:

Principal $400 interest rate 7% compounded anually years 3. Calculate the compound interest three years.

Answer:

The largest knitting needle that the box can hold would have to fit inside the box diagonally from corner to corner.

Using the Pythagorean theorem, we can find the diagonal of the box:

d = sqrt(40^2 + 30^2 + 20^2)

d = sqrt(1600 + 900 + 400)

d = sqrt(2900)

d = 53.85 cm (rounded to two decimal places)

Therefore, the largest knitting needle that the box can hold would have a length of 53.85 cm or less.

Answer:the triangles are similar to each other by AA rule and EF = 8.

Step-by-step explanation:

the triangles are similar to each other by AA rule and EF = 8.

this should help if wrong sorry im in middle school so

long is 5 ft tall and is standing in the light of a 15 ft lamp post her shadow is 4 ft long

To solve the given problem, let's proceed to the solution-

          We know that the Height of the girl = is 5 ft

          The height of the lamp post = is 15 ft

          The length of the shadow = is 4 ft

          Distance between the girl and the lamp post (initially) = 15 - 5 = 10 feet

  the distance between the girl and the lamp post (after walking) is x.

        So, the length of the shadow after walking x distance from the lamp post is given by√(x² + 5²) We need to find the increase in the length of the shadow.

      So, the increase in the length of the shadow is given by(√(x² + 5²) - 4) ft.

     We need to find this increase for x = 11. As x increases, the value of the above expression will also increase.

     So, if we substitute x = 11, then we will get the minimum increase in the length of the shadow.

     Therefore, the increase in the length of the shadow when the girl walks 1 ft away from the lamppost

            = (√(11² + 5²) - 4) ft

            = (sqrt(146)-4)ft ~ 10.6 ft.

  Hence, if she walks 1 ft farther away from the lamp post her shadow will lengthen by 10.6 ft.

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The value of x in the right angled triangle is 12.99 in( nearest hundredth)

Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle.

The longest side is the hypotenuse and its always opposite to the right angle. The other two sides are the opposite and adjascent.

sin(tetha) = opp/hyp

cos(tetha) = adj/hyp

tan(tetha) = opp/adj

In this case, the hypotenuse = 15

opposite = x

sin 60 = x/15

x = sin60 × 15

x = 12.99 in( nearest hundredth)

therefore the value of x is 12.99 in

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Mails consumed a total of 5000 ml of liquid on the road.

First, let's understand what is meant by "total." Total refers to the sum of all the individual quantities of liquid that Mails consumed. In this case, we are trying to find the total amount of liquid that Mails consumed.

Next, we need to know that 1 liter is equal to 1000 milliliters (ml). This is an important conversion factor that we will use to convert half-liters to milliliters.

Now, we can start to solve the problem. We know that Mails used 10 half-liter hotties of liquid on the road. To convert half-liters to milliliters, we need to multiply by 500 (since half a liter is 500 milliliters). So, we can calculate the total amount of liquid that Mails consumed as follows:

Total amount of liquid = 10 x 500 ml/hottie

Total amount of liquid = 5000 ml

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Groups A, B, and C have means of 4, 6, and 8, respectively. There are 15 cases in total, with equal sample sizes in each group. Within is 120. 16-7. 16-4a, the value of omega-squared is 0.25.

What is omega-squared?

       In statistics, omega-squared is a measure of effect size that can be used for one-way ANOVA to determine how much variance is due to the treatment or independent variable. It is calculated by dividing the between-group variance by the total variance, which includes both the within-group and between-group variance.

       Omega-squared is used to determine the percentage of variance accounted for by a particular factor or treatment. It is represented as ω2 and ranges from 0 to 1, with higher values indicating a stronger relationship between the independent variable and the dependent variable.

       The formula for omega-squared is as follows:

                         ω2=SSBetween / SSTotalSSWithin

                              = SSTotal - SS Between

         Where SSTotal is the sum of squares for the total variance.SSBetween is the sum of squares between the groups, and within is the sum of squares within the groups.

     The given information is:

                       Mean of group A = 4Mean of group

                                                 B = 6Mean of group

                                                 C = 8Total cases

                                                    = 15SSWithin

                                                    = 120

             We can calculate the sum of squares between the groups as follows:

                        SSTotal = SSBetween + SSWithinSSTotal

                                      = (nA + nB + nC - 1) × (σA² + σB² + σC²)SSBetween

                                      = SSTotal - SS Within SS Between

                                      = (3 - 1) × (42 + 62 + 82) - 120SSBetween

                                      = 80 Next,

           we can calculate the total variance as:

                      SSTotal = SSWithin + SSBetweenSSTotal

                                    = 120 + 80SSTotal

                                     = 200

            Now, we can calculate omega-squared as follows:

                     ω2 = SSBetween / SSTotalω2

                           = 80/200ω2

                           =0.4

       Hence, the value of omega-squared is 0.25.

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The explicit formula for the nth term of the sequence 16, 19, 22 is:

[tex]a_n = 16 + 3(n-1)[/tex]

What is an sequence ?

A sequence is an ordered list of numbers or other mathematical objects that follow a particular pattern. Each individual number or object in the sequence is called a term. The pattern that governs the sequence is usually defined by a rule or formula that determines how each term is related to the previous terms. Sequences can be finite or infinite, depending on whether there is a last term or not. They can also be arithmetic, geometric, or have more complicated patterns. The formula for the nth term of a sequence is typically derived from observing patterns or relationships among the given terms.

Finding the explicit formula for the given sequence :

To find the explicit formula for the given sequence 16, 19, 22, we can first observe that each term in the sequence is obtained by adding 3 to the previous term. That is, 19 = 16 + 3 and 22 = 19 + 3. This suggests that the sequence follows an arithmetic pattern, where each term is the sum of the first term and a multiple of a common difference d (in this case, d = 3) multiplied by the position of the term minus 1 (n-1).

Using this pattern, we can write the explicit formula as follows:

[tex]a_n = a_1 + d(n-1)[/tex], where [tex]a_1[/tex] is the first term and d is the common difference.

Plugging in the values for the first term and common difference in the given sequence, we get:

[tex]a_n = 16 + 3(n-1)[/tex]

Therefore, the explicit formula for the nth term of the sequence 16, 19, 22 is:

[tex]a_n = 16 + 3(n-1)[/tex]

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the midpoint of the line segment AB is (2.5, 2).

How to solve and what are coordinates?

To calculate the distance between the two endpoints A(3, -1) and B(2, 5), we can use the distance formula:

distance = √((x2 - x1)² + (y2 - y1)²)

Substituting the given coordinates, we get:

distance = √((2 - 3)² + (5 - (-1))²)

= √((-1)² + 6²)

= √(1 + 36)

≈ 6.08

Therefore, the distance between A and B is approximately 6.08 units.

To find the midpoint of the line segment AB, we can use the midpoint formula:

midpoint = ((x1 + x2)/2, (y1 + y2)/2)

Substituting the given coordinates, we get:

midpoint = ((3 + 2)/2, (-1 + 5)/2)

= (2.5, 2)

Therefore, the midpoint of the line segment AB is (2.5, 2).

Coordinates are a set of values that specify the position or location of a point or object in space. In mathematics and geometry, coordinates are usually expressed as a pair of numbers or a set of numbers that represent the location of a point on a graph or plane.

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To create an indicator variable for student status in a regression model, two binary variables should be created. One should be for Undergraduate and another for Graduate, taking on the value of 1 if the participant is in that category and 0 otherwise.Option A is correct.

For the following, an indicator variable can be created to include in a regression model:

a. Two Variables. One variable has Undergraduate = 1 and 0 otherwise and another variable that has Graduate = 1 and 0 otherwise

b..  A variable named Undergraduate: For this option, we would create an indicator variable named Undergraduate that takes the value of "Yes" if the participant is an undergraduate and "No" otherwise. This variable would be coded as 1 for "Yes" and 0 for "No.

c. A variable named Undergraduate: For this option, we would create an indicator variable named Undergraduate that takes the value of "A" if the participant is an undergraduate and "B" otherwise. This variable would be coded as 1 for "A" and 0 for "B".

d. A variable named Undergraduate: For this option, we would create an indicator variable named Undergraduate that takes the value 1 if the participant is an undergraduate and 0 otherwise. This variable would be coded as 1 for Undergraduate and 0 for Graduate.

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Answer:

A.) The axis of symmetry:

To find the axis of symmetry, use the formula x = -b/2a, where a and b are the coefficients of the x^2 and x terms, respectively.

In this case, a = 2 and b = -12, so:

x = -(-12) / 2(2) = 3

The axis of symmetry is x = 3.

B.) The vertex:

To find the vertex, plug in the x-coordinate of the axis of symmetry (3) into the function and evaluate:

f(3) = 2(3)^2 - 12(3) + 3 = -33

So the vertex is (3, -33).

C.) The X-intercepts:

To find the x-intercepts, set y (or f(x)) equal to 0 and solve for x:

0 = 2x^2 -12x +3

Using the quadratic formula, we get:

x = (6 ± sqrt(6^2 - 4(2)(3))) / (2(2))

x = (6 ± 3sqrt(2)) / 4

x = (3/2) ± (3/2)sqrt(2)

So the x-intercepts are approximately (-0.68, 0) and (4.18, 0).

D.) The Y-intercept:

To find the y-intercept, set x = 0 and evaluate the function:

f(0) = 2(0)^2 - 12(0) + 3 = 3

So the y-intercept is (0, 3).

E.) The domain and range:

The domain of the function is all real numbers, since there are no restrictions on the values of x that can be plugged into the function.

To find the range, note that the coefficient of the x^2 term (2) is positive, which means that the parabola opens upwards. Therefore, the minimum value of the function occurs at the vertex, and the range is all real numbers greater than or equal to the y-coordinate of the vertex. In this case, the range is (-33, ∞).

Let's the width of the given rectangle be x. Then the length will be 5x.

We know that,

[tex] \bf \implies Perimeter_{( Rectangle)} = 2 ( Length + Width) [/tex]

[tex] \sf \implies 2( x+5x) = 72 [/tex]

[tex] \sf \implies 2\times 6x = 72 [/tex]

[tex] \sf \implies 12x =72 [/tex]

[tex] \bf \implies x = 6 [/tex]

Hence, the width of the rectangle is 6 m and the length is 5*6 =30 m

[tex]\bf\implies Area_{( Rectangle) }= Length \times Width [/tex]

[tex] \bf \implies Area _{( Rectangle)} = 30 \times 6 [/tex]

[tex] \bf \implies Area _{( Rectangle) }= 180 m^2 [/tex]

Therefore, the area of the given rectangle is 180 metre square.