Determine the maximum rate of change of f at the given point P and the direction in which it occurs (a) f(x,y) = sin(xy), P(1,0) (b) f(x,y,z) = P(8,1.3)

Answer:

x = 1/2, -4/3

Step-by-step explanation:

(2x - 1)(3x + 4) = 0

We know any number multiplied by 0 will be equal 0.

2x - 1 = 0

2x = 1

x = 1/2

3x + 4 = 0

3x = -4

x = -4/3

So, x = 1/2, -4/3

[tex]2x(3x + 4) - 1(3x + 4) = 0[/tex]

[tex]6 {x}^{2} + 8x - 3x - 4 = 0[/tex]

[tex]6 {x}^{2} + 5x - 4 = 0[/tex]

[tex]6 {x}^{2} + 5x = 4[/tex]

The exact value of sin 45° is [tex]\dfrac{\sqrt{2} }{2}[/tex]

Since we have given that

[tex]\text{sin} \ 45^\circ[/tex]

We need to find the exact value of sin 45°.

From the trigonometric table,

[tex]\text{sin} \ 45^\circ=\dfrac{1}{\sqrt{2}}[/tex]

We need to write it as a simplified fraction,

So, for this, we will rationalize the denominator:

[tex]\dfrac{1}{\sqrt{2}}\times\dfrac{\sqrt{2} }{\sqrt{2}}[/tex]

[tex]=\dfrac{\sqrt{2} }{2}[/tex]

Hence, the exact value of sin 45° is [tex]\dfrac{\sqrt{2} }{2}[/tex]

Answer: 1 divided by the square root of 2

Step-by-step explanation:

Let's set up an example, if the angle is forty five degrees, and the opposite length is 1, we can solve this as sin to get to the hypotenuse,

1. sin(45) = 1/hyp

2. sin(45) times hyp = 1

3. hyp = sin(45)/1

If we take any answer and put it over the hypotenuse as sin, we can see that it is going to end up as 1/√2, or 0.707

I did 1 because you are just asking for sin(45).

The quantity that gives maximum profit is 1,750 units.

To find the quantity that gives maximum profit, we first need to determine the profit function.

Profit = Total Revenue - Total Cost

Total Revenue is given by the price (p) times the quantity (q):

TR = pq

Since the corporation has monopolistic control, it can set the price to maximize profit. We can use the demand function

to find the price that will maximize profit:

Q = 180,000 - 50p

Solving for p, we get:

p = 3,600 - 0.02Q

Now we can substitute this into the profit equation:

Profit =[tex](3,600 - 0.02Q)Q - (900 + 100Q + Q^2)[/tex]

Simplifying:

Profit = [tex]-Q^2 + 3,500Q - 900[/tex]

To find the quantity that gives maximum profit, we can take the derivative of the profit function with respect to Q and

set it equal to zero:

[tex]d/dQ (-Q^2 + 3,500Q - 900) = 0[/tex]

-2Q + 3,500 = 0

Q = 1,750

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Laura is paying each month:

extra payment = $550 - monthly payment

To calculate how much extra Laura is paying each month, we first need to calculate the total cost of her car financing. Here are the steps:

Calculate the trade-in value of Laura's old car. We don't have the exact value, but we know that she received 85% of the trade-in value listed below, so we can set up an equation:

0.85 * trade-in value = value Laura received

Solving for the trade-in value, we get:

trade-in value = value Laura received / 0.85

Add the trade-in value to the list price of the new car to get the total cost before taxes and fees:

total cost before taxes and fees = $19,858 + trade-in value

Add the sales tax, registration fee, and documentation fee to get the total cost of the car financing:

total cost = (1 + 0.0927) * total cost before taxes and fees + $988 + $77

Calculate the monthly payment using the formula for a loan with monthly compounding:

monthly payment = (principal * monthly interest rate) / (1 - (1 + monthly interest rate)^(-number of months))

We know that Laura is financing the rest of the cost after her trade-in value, so:

principal = total cost - value Laura received

monthly interest rate = 0.095 / 12

number of months = 48 (since it's a four-year financing)

Substituting these values into the formula, we get:

monthly payment = ($19,858 + trade-in value - value Laura received) * 0.007916 / [tex](1 - (1 + 0.007916)^{(-48)})[/tex]

Now that we have the total monthly payment, we can calculate how much extra Laura is paying each month:

extra payment = $550 - monthly payment

Note that this assumes that Laura doesn't have any other fees or interest charges on her car financing, such as late payment fees or penalties for paying off the loan early. If there are any additional fees or charges, the calculation may be different.

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To calculate ∑ (-1)^k pi^2k/2k, we can use the power series expansion for the cosine function:

cos(x) = ∑ (-1)^n x^(2n) / (2n)!

We can substitute pi^2 for x in this formula to get:

cos(pi^2) = ∑ (-1)^n (pi^2)^(2n) / (2n)!
= ∑ (-1)^n pi^(4n) / (2n)!

Now we can compare this to the original series we want to evaluate:

∑ (-1)^k pi^2k/2k = ∑ (-1)^n pi^(2n) / (2n)

We notice that the powers of pi in the two series match up, but the coefficients are different. However, we can use the identity cos(pi^2) = (-1)^n to rewrite the series we want to evaluate as:

∑ (-1)^n pi^(2n) / (2n) = ∑ (-1)^n pi^(4n) / (2n)! * (2n) / pi^2n

= pi^2 / 2 * ∑ (-1)^n pi^(4n) / (2n)!

Now we can substitute our expression for cos(pi^2) into this equation to get:

∑ (-1)^k pi^2k/2k = pi^2 / 2 * cos(pi^2)

= pi^2 / 2 * (-1)^n

Therefore, the value of the series is (-1)^n * pi^2 / 2.
To calculate the sum ∑ (-1)^k (pi^2k)/(2k), it is important to recognize that this is an alternating series with a general term given by a_k = (-1)^k (pi^2k)/(2k).

However, the question seems incomplete, as there is no specified range for the sum (i.e., the values of k). If you provide the range of k for which this sum is to be calculated, I would be glad to help you further.

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Calculated t-value of 6.93 is greater than the critical value, we reject the null hypothesis and conclude that there is strong evidence that the mean amount of nitrogen in distributor A's fertilizer is significantly greater than the mean amount of nitrogen in distributor B's fertilizer.

Since we do not have the population standard deviation, we will need to use the t-distribution for our hypothesis test. We are interested in testing whether the mean amount of nitrogen in distributor A's fertilizer is significantly greater than the mean amount of nitrogen in distributor B's fertilizer.

Let μA be the true mean amount of nitrogen in distributor A's fertilizer and μB be the true mean amount of nitrogen in distributor B's fertilizer. Our null hypothesis is:

H0: μA - μB ≤ 0

The alternative hypothesis is:

Ha: μA - μB > 0

We will use a one-tailed test with a significance level of 0.05. Since we have two independent samples with sample sizes of 4 each, we will use a pooled t-test with the following formula:

t = ([tex]\bar{X1}[/tex] - [tex]\bar{X2}[/tex] - D) / (sP * √(2/n))

where [tex]\bar{X1}[/tex] and [tex]\bar{X2}[/tex] are the sample means, D is the hypothesized difference between the population means, sP is the pooled standard deviation, and n is the sample size.

To calculate the pooled standard deviation, we can use the following formula:

sP = √(((n1-1)*s1² + (n2-1)*s2²) / (n1+n2-2))

where n1 and n2 are the sample sizes, and s1 and s2 are the sample standard deviations.

Plugging in the given values, we get:

[tex]\bar{X1}[/tex] = 21, [tex]\bar{X2}[/tex] = 16

s1 = s2 = 1.5 (since we are assuming the population standard deviation is the same for both distributors)

n1 = n2 = 4

D = 0 (since the null hypothesis is that there is no difference in the means)

sP = √(((4-1)*1.5² + (4-1)*1.5²) / (4+4-2)) = 1.5

Using these values, we get:

t = (21 - 16 - 0) / (1.5 * √(2/4)) = 6.93

Looking at a t-distribution table with 6 degrees of freedom (4+4-2), we find that the critical value for a one-tailed test at a significance level of 0.05 is approximately 1.943. Since our calculated t-value of 6.93 is greater than the critical value, we reject the null hypothesis and conclude that there is strong evidence that the mean amount of nitrogen in distributor A's fertilizer is significantly greater than the mean amount of nitrogen in distributor B's fertilizer.

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Complete Question:

A farmer uses a lot of fertilizer to grow his crops. The farmer's manager thinks fertilizer products from distributor A contain more of the nitrogen that his plants need than distributor B's fertilizer does. He takes two independent samples of four batches of fertilizer from each distributor and measures the amount of nitrogen in each batch. Fertilizer from distributor A contained 23 pounds per batch and fertilizer from distributor B contained 18 pounds per batch. Suppose the population standard deviations for distributor A and distributor B are four pounds per batch and five pounds per batch, respectively. Assume the distribution of nitrogen in fertilizer is normally distributed. Let ?1 and ?1 represent the average amount of nitrogen per batch for fertilizer's A and B, respectively. Calculate the value of the test statistic

The 90% confidence interval of the new product has the potential to make the company an additional $3.8 million is (2.81, 4.79), option B.

Confidence intervals quantify how confident or uncertain a sampling technique is. They can take any number of probability thresholds, with a 95% or 99% confidence level being the most popular. The calculation of confidence intervals is done using statistical techniques like the t-test.

[tex]\bar x = $3.8[/tex]

sample standard deviation = s =1.7

sample size = n =10

Degrees of freedom = df = n - 1 = 10 - 1 = 9

At 90% confidence level

[tex]\alpha[/tex] = 1 - 90%

[tex]\alpha[/tex] =1 - 0.90 =0.1

[tex]\alpha/2[/tex] = 0.05

[tex]t\alpha/2[/tex] ,df = t0.05,9 =1.833

At 90% confidence level, the critical value is t = 1.833

The 90% confidence interval is:

[tex]\bar x \pm t*\frac{s}{\sqrt{n} }[/tex]

[tex]=3.8\pm 1.833*\frac{1.7}{\sqrt{10}}\\\\=3.8\pm 0.99[/tex]

=(2.81,4.79).

Therefore, 90% confidence interval estimate of the population mean is, (2.81,4.79).

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Complete question:

Market research indicates that a new product has the potential to make the company an additional $3.8 million, with a standard deviation of $1.7 million. If this estimate was based on a sample of 10 customers, what would be the 90% confidence interval? (2.76, 4.84) O (2.81, 4.79) (2.11, 5.56) (3.06, 4.54)

We know that the function (f+ g)(x) = 13x^2 + 8x + 2 and (f+ g)(-2) = 38.

Hi! I'd be happy to help you with your question.

Given the functions f(x) = 9x^2 + 8x + 2 and g(x) = 4x^2, we need to find (f+ g)(x) and (f+ g)(-2).

To find (f+ g)(x), simply add the functions f(x) and g(x) together:

(f+ g)(x) = f(x) + g(x) = (9x^2 + 8x + 2) + (4x^2) = 13x^2 + 8x + 2

Now, we need to find (f+ g)(-2) by substituting -2 for x in the combined function:

(f+ g)(-2) = 13(-2)^2 + 8(-2) + 2 = 13(4) - 16 + 2 = 52 - 14 = 38

So, (f+ g)(x) = 13x^2 + 8x + 2 and (f+ g)(-2) = 38.

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Alleen's mom is not correct because if her friend reads 20 pages, she can read 30 pages in the same time period.

To answer this question, we need to use the terms "ratio" and "proportion". The given ratio of Alleen's reading speed to her friend's speed is 1.5:1. If Alleen's friend reads 20 pages, we can use proportion to find how many pages Alleen can read:

1.5 / 1 = x / 20

To solve for x (the number of pages Alleen reads), we can cross-multiply:

1.5 * 20 = 1 * x
30 = x

So, if Alleen's friend reads 20 pages, Alleen can read 30 pages in the same time period.

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a) The characteristic equation is r^2 + 11r + 28 = 0.

b) The associated homogeneous equation are y1(x) = c1e^(-4x) and y2(x) = c2e^(-7x).

c) The form of the particular solution is y_p(x) = Ae^(5x) + Bx + C.

d) By solving the system of equations, it gives A = 1/28, B = 1/28, and C = -211/196.

e) The general solution is y(x) = c1e^(-4x) + c2e^(-7x) + (1/28)e^(5x) + (1/28)x - 211/196.

a) The characteristic equation of the associated homogeneous equation is r^2 + 11r + 28 = 0.

b) Factoring the characteristic equation gives (r + 4)(r + 7) = 0, so the solutions to the associated homogeneous equation are y1(x) = c1e^(-4x) and y2(x) = c2e^(-7x).

c) The form of the particular solution is y_p(x) = Ae^(5x) + Bx + C. Taking the first and second derivatives of y_p(x) gives y_p'(x) = 5A + B and y_p''(x) = 0.

d) Substituting y_p(x), y_p'(x), and y_p''(x) into the original nonhomogeneous equation gives:

0 + 11(5A + B) + 28(Ae^(5x) + Bx + C) = e^(5x) + x + 4

Simplifying this equation gives:

(28A)e^(5x) + (28B)x + 11(5A) + 11B + 28C = e^(5x) + x + 4

Comparing coefficients gives the system of equations:

28A = 1

28B = 1

11(5A) + 11B + 28C = 4

Solving this system of equations gives A = 1/28, B = 1/28, and C = -211/196.

e) The general solution to the nonhomogeneous equation is y(x) = y_h(x) + y_p(x), where y_h(x) = c1e^(-4x) + c2e^(-7x) and y_p(x) = (1/28)e^(5x) + (1/28)x - 211/196. Therefore, the general solution is:

y(x) = c1e^(-4x) + c2e^(-7x) + (1/28)e^(5x) + (1/28)x - 211/196.

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The height of the lean-to on its vertical side is 8 feet.

To find the height of the lean-to on its vertical side, we need to use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In this case, the vertical side is the hypotenuse, and the length and width are the other two sides.

So, we have:

[tex]height^2 = hypotenuse^2 - width^2[/tex]

We know the length of the roof (the hypotenuse) is 17 feet, and the width is 15 feet. So we can plug these values into the equation and solve for the height:

[tex]height^2 = 17^2 - 15^2\\height^2 = 289 - 225\\height^2 = 64\\height = 8[/tex]

Therefore, the height of the lean-to on its vertical side is 8 feet.

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Answer:

gf(x) = g(f(x)) = (x+4)(2x/x-2) = 2x^2 + 8x - 8

The average change in temperature per hour during the temperature drop from 48°F to -16°F in Sioux Falls, South Dakota on February 6, 1995, was 8°F per hour.

On February 6, 1995, the temperature in Sioux Falls, South Dakota dropped dramatically from 48°F to -16°F in just eight hours. To calculate the average change in temperature per hour, we can use the formula:

Average Change in Temperature per Hour = (Change in Temperature) ÷ (Time)

Using this formula, we can calculate the average change in temperature per hour in Sioux Falls as follows:

Average Change in Temperature per Hour = (48°F - (-16°F)) ÷ 8 hours

Average Change in Temperature per Hour = 64°F ÷ 8 hours

Average Change in Temperature per Hour = 8°F per hour

Therefore, the average change in temperature per hour during that eight-hour period in Sioux Falls, South Dakota was 8°F. This rapid and significant change in temperature was likely due to a strong cold front moving through the area.

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The point on the curve y = -4x² - 2x - 11 where the tangent is parallel to the line 8x - 3 is (-1, -13).

To find the points on the curve where the tangent is parallel to the line, we need to find where the derivative of the curve is equal to the slope of the line.

The given curve is: y = -4x² - 2x - 11

The derivative of this curve is: y' = -8x - 2

The slope of the given line is: 8

We want to find the points where the derivative of the curve is equal to the slope of the line:

-8x - 2 = 8

Solving for x, we get:

x = -1

Now, we can plug this value of x back into the original equation to find the corresponding value of y:

y = -4(-1)² - 2(-1) - 11

y = -13

Therefore, the point on the curve where the tangent is parallel to the line 8x - 3 is (-1, -13).

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Note that the price to be paid for each bond if they are purchased today a.

ABC: $1047.50

XYZ: $1005.00 (Option A)

The formula to determine the price to pay for a bond,  is ...

Price = (Annual Interest Payment) / (Current Yield)

where Annual Interest Payment = (Coupon Rate / 100) x Face Value, and

Current Yield = (Annual Interest Payment / Price) x 100.

Using the given information, we can calculate the price to pay for each bond

For ABC bond

Annual Interest Payment

= (7.5 / 100) x $1000 = $75

Current Yield

= (Annual Interest Payment / Price) x 100 = (75 / $1042.50) x 100

= 7.2%

Price = (Annual Interest Payment) / (Current Yield)

= $75 / (7.2/100)

= $1041.67

So .... the price to pay for the ABC bond is approximately $1041.67.

For XYZ bond

Annual Interest Payment

= (8.4 / 100) x $1000

= $84

Current Yield

= (Annual Interest Payment / Price) x 100

= (84 / $1003.125) x 100

= 8.37%

Price = (Annual Interest Payment) / (Current Yield)

= $84 / (8.37/100)

= $1003.84

So, the price to pay for the XYZ bond is approximately $1003.84.

So, the closest option to the calculated prices is:

a. ABC: $1047.50

XYZ: $1,005.00

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Check the picture below.

Among rectangles, rhombuses, and squares, all three of these shapes are parallelograms that have specific properties.

A rectangle is a parallelogram with four right angles. Its opposite sides are equal and parallel, and it has diagonals that are equal in length and bisect each other.

A rhombus, on the other hand, is a parallelogram with all four sides being equal in length. Like a rectangle, its opposite sides are parallel, but it does not necessarily have right angles. The diagonals of a rhombus are perpendicular bisectors of each other, meaning they intersect at a 90-degree angle and divide each other into two equal parts.

Finally, a square is a special type of parallelogram that combines the properties of both rectangles and rhombuses. It has four equal sides and four right angles, making it a unique shape. The diagonals of a square are equal in length, bisect each other, and are also perpendicular bisectors.

In conclusion, rectangles, rhombuses, and squares are all parallelograms with distinct properties. Rectangles have right angles and equal opposite sides, rhombuses have equal sides and diagonals that are perpendicular bisectors, and squares possess all the properties of both rectangles and rhombuses.

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The probability of being assigned to a desk in the front row exactly four times over the course of seven days is approximately 0.008, or 0.8%.

There are a total of 16 desks in the classroom, arranged in 4 rows and 4 columns. The probability of being assigned to a desk in the front row is 4/16 = 1/4, since there are 4 desks in the front row.

To calculate the probability of being assigned to a front-row desk exactly 4 times over the course of 7 days, we can use the binomial probability formula:

P(X = k) = (n choose k) * p^k * (1 - p)^(n-k)

where X is the random variable representing the number of times you are assigned to a front-row desk, n is the number of trials (in this case, 7), k is the number of successes (being assigned to a front-row desk), p is the probability of success on each trial (1/4), and (n choose k) represents the number of ways to choose k successes out of n trials, which is given by the binomial coefficient formula:

(n choose k) = n! / (k! * (n-k)!)

where ! represents the factorial function.

Using this formula, we get:

P(X = 4) = (7 choose 4) * (1/4)^4 * (3/4)^3

P(X = 4) = (35) * (1/256) * (27/64)

P(X = 4) ≈ 0.008

Therefore, the probability of being assigned to a desk in the front row exactly four times over the course of seven days is approximately 0.008, or 0.8%.

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Answer:

4x² + 24x + 35

Step-by-step explanation:

Total area = (7 + 2x)(5 + 2x)

= 35 + 10x + 14x + 4x²

= 4x² + 24x + 35

Answer:

7776 outcomes

Step-by-step explanation:

To get the total number of outcomes we multiply the total number of possibilities for each roll. Since there are 5 rolls, the total number of outcomes will be:

6 x 6 x 6 x 6 x 6 = 7776 outcomes

The probability of a bulb lasting for at most 624 hours is 0.9861, rounded to four decimal places.

The standard deviation of the lifetime is a measure of how spread out the lifetimes are. In other words, it tells us how much the lifetimes of bulbs vary from the mean. In this case, the standard deviation of the lifetime is 20 hours.

Now, let's get to the question at hand. We want to find the probability of a bulb lasting for at most 624 hours. To do this, we need to use the properties of the normal distribution.

First, we need to calculate the z-score, which tells us how many standard deviations a value is from the mean. We can use the formula z = (x - mu) / sigma, where x is the value we are interested in, mu is the mean, and sigma is the standard deviation. In this case, x = 624, mu = 580, and sigma = 20.

Plugging these values into the formula, we get z = (624 - 580) / 20 = 2.2.

Next, we need to find the probability of a bulb lasting for at most 624 hours, which is the same as finding the area under the normal curve to the left of z = 2.2. We can use a standard normal distribution table or a calculator to find this probability.

Using a calculator, we can use the normal cdf function with the values -9999 (a very large negative number) and 2.2 to find the probability. This gives us a probability of 0.9861.

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The proportion of students from a random sample of 500 fail to reject the null hypothesis and can not support Mr. Smith's claim as per the data.

Percent of students claim they have at least two cell phones = 20%

Sample size = 500

Significance level α = 0. 05

This is a hypothesis testing problem with the following hypotheses,

Null hypothesis (H₀),

The proportion of students who have at least two cell phones is 0.20.

Alternative hypothesis (Hₐ),

The proportion of students who have at least two cell phones is greater than 0.20.

Use a one-tailed z-test for proportions to test the null hypothesis at a significance level of α = 0.05.

The test statistic is calculated as,

z = (p₁ - p₀) / √(p₀(1-p₀)/n)

where p₁ is the sample proportion,

p₀ is the null hypothesis proportion,

and n is the sample size.

Using the values in the problem, we get,

p₁ = 88/500

   = 0.176

p₀ = 0.20

n = 500

z = (0.176 - 0.20) / √(0.20(1-0.20)/500)

  = -1.34

Using a standard normal distribution table,

the p-value for z = -1.34 is approximately 0.0901.

Since the p-value (0.0901) is slightly greater than the significance level (0.05),

Fail to reject the null hypothesis.

Do not have sufficient evidence to conclude that the proportion of students who have at least two cell phones is greater than 0.20.

Therefore, cannot support Mr. Smith's claim based on the given data as fail to reject the null hypothesis.

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Option B is correct. The relationship is: H0 = 0 there is no linear association between calories and sodium content H1  ≠ 0 there is a linear association between colones and sodium content

The test statistic is 3.75

The test statistics can be gotten from the data that we already have available in this question

The coefficient is given as 2.235

The Standard error of the coefficient is given as 0.596

The formula used is given as

Such that t = coefficient /  Standard error

where the coefficient = 2.235

The standard error = 0.596

Then when we apply the formula we have

2.235 / 0.596

t statistic = 3.75

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Answer:

2

Step-by-step explanation:

The K and N are the trapeziums and the two lines opposite to them go in a parallel line

Kaylie earned $693 in interest on her savings account.

Kaylie deposited $900 in a savings account and earned 11% interest on it for 7 years. After the 7-year period, Kaylie earned $693 in interest, which was compounded annually.

This brought the total amount of money in her savings account to $1,593. The power of compounding interest in savings accounts can help grow your money significantly over time.

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The value of w in the expression is 10.2

The expression is given as, we are required to calculate the value of w

4(-4.6 +w)= 22.24

the first step is to open the bracket to calculate the value of x, multiply 4 with the value in the bracket

-18.4 + 4w= 22.24

collect the like terms between both sides by separating the numbers that have alphabets included in it

4w= 22.24 + 18.4

4w= 40.8

Divide by the coefficient of w which is 4

4w/4= 40.8/4

w= 40.8/4

w= 10.2

Hence the value of w in the expression is 10.2

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Answer:

Step-by-step explanation:

The statement that correctly describes the solution to this system of equations 4x+2y=6 and 4x+2y=4 is "There is no solutions". The correct option is A.

The given system of equations is 4x + 2y = 6 and 4x + 2y = 4.

On comparing the two equations, we notice that the left-hand side of both the equations is the same. However, the right-hand side of the two equations is different. This implies that the lines represented by the two equations are parallel to each other, since they have the same slope but different y-intercepts.

If two lines are parallel, they will never intersect. In this case, since the two equations represent two parallel lines, there is no point of intersection between them. Therefore, the system of equations has no solution.

Hence, the correct answer is A. There is no solution to this system of equations.

In summary, the given system of equations cannot be satisfied simultaneously, since the lines represented by the two equations are parallel to each other and hence do not intersect. Therefore, the system of equations has no solution.

To know more about equations and solutions, refer to the link below:

https://brainly.com/question/30127282#

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