Solve the system of linear equations by elimination

4x+6y=48
3x + 7y=51

The probability of the first toy being defective is 1/3 and the probability of both toys being defective is 1/15.

Hence, option 'a'.

Use the concept of probability defined as:

Probability is equal to the percentage of positive outcomes compared to all outcomes for the occurrence of an event.

Given that,

Mrs. Jones buys her son two toys.

The probability that the initial plaything is broken =  1/3.

The probability that the second toy has a problem.,

Also given the original toy is defective, is 1/5.

The goal is to calculate the probability that both toys are defective.

To calculate the probability that both toys are defective:

Multiply the individual probabilities.

Given that the probability of the first toy being defective = 1/3,

The probability of the second toy being defective given that the first toy is defective= 1/5,

Calculate the probability that,

Both toys have flaws, multiplying these probabilities together:

(1/3) x (1/5) = 1/15

Hence,

The probability that both toys are defective is [tex]1/15[/tex] which is option 'a'.

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The complete question is:

Mrs. Jones buys two toys for her son. the probability that the first toy is defective is 1/3, and the probability that the second toy is defective given /that the first toy is defective is 1/5. what is the probability that both toys are defective?

a.  1/15

b. 1/4

c. 1/8

d. 3/5

Thus, the percentage raise received on the family's $88,000 yearly income is 9.4%.

The difference in between final value and the starting value, stated as a percentage, is known as a percentage increase.

The base amount still determines whether a percentage rise or drop by a given percentage occurs. The absolute value change also changes if the basic amount does.

Hence, although the percentage rise or reduction is the same in this instance, the absolute increase is different.

Given data:

From the table, compound interest for the yearly inflation rate of 3% is 1.09417024.

Thus,

amount after compounding:

A = $88,000  * 1.09417024.

A = 96286.98112

A = $96286.98

percentage raise = (96286.98 - 88,000 )/ 88,000

percentage raise = 0.094 * 100

percentage raise = 9.4%

Thus, the percentage raise received on the family's $88,000 yearly income is 9.4%.

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Answer:

Step-by-step explanation:

Unfortunately, I cannot answer this question without additional information about the distances traveled and the time taken by Reiko to travel from point A to point B and back to point A.

The speed at which Reiko traveled is calculated as distance divided by time. Therefore, we need to know both the distance and time for each leg of the journey to determine the speed.

Without this information, it is not possible to determine whether Reiko traveled from A to B at a speed greater than 40 miles per hour.

The greatest number of teams the director could make is 4, and there will be 3 girls on each team.

Since the director wants to make teams with an equal number of boys and girls, the number of teams must be a factor of both 16 and 12. The common factors of 16 and 12 are 1, 2, 4, and 8. Since the director wants to make as many teams as possible, the greatest number of teams is 4.

Each team will have 4 boys and 3 girls, so the total number of girls needed is 4 x 3 = 12. Since there are 12 girls in the camp, there will be 12/4 = 3 girls on each team. Therefore, the greatest number of teams the director could make is 4, and there will be 3 girls on each team.

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Maria jumped 6 inches farther than Nate.

To see why, we can subtract Nate's jump height from Maria's jump height:

32 - 26 = 6

So Maria jumped 6 inches farther than Nate did.

To represent this problem mathematically, we can use the equation:

Maria's jump height - Nate's jump height = the difference in their jump heights

Or, using variables:

M - N = D

Where M represents Maria's jump height, N represents Nate's jump height, and D represents the difference between their jump heights. Plugging in the numbers from the problem, we get:

32 - 26 = D

Simplifying, we get:

6 = D

So D, the difference between their jump heights, is 6 inches.

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If the CPI was 305 in 1995, the price of jeans that cost $14.99 in 1973 would be approximately $47.05 in 1995 after adjusting for inflation.

To find the price of jeans in 1995, we first need to adjust the 1973 price for inflation using the Consumer Price Index (CPI). CPI measures the average change in prices of goods and services over time, so it can help us compare prices from different years.

First, we need to calculate the inflation rate between 1973 and 1995. We can do this by dividing the CPI in 1995 (305) by the CPI in 1973 (135):

Inflation rate = (305 / 135) * 100% = 226.67%

This means that prices in 1995 were about 2.27 times higher than in 1973. Now, we can apply this inflation rate to the price of jeans in 1973:

Price in 1995 = Price in 1973 * (1 + inflation rate)

Price in 1995 = $14.99 * (1 + 2.2667) = $47.05

Therefore, if the CPI was 305 in 1995, the price of jeans that cost $14.99 in 1973 would be approximately $47.05 in 1995 after adjusting for inflation. This calculation helps to compare the cost of goods across different time periods by taking inflation into account, thus giving a better understanding of the changes in purchasing power over time.

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To find the linearization L(x) of the function f(x) = 7cos(x) at a = 3.14159265359, we'll use the formula:

L(x) = f(a) + f'(a)(x - a)

where f'(x) is the derivative of f(x) with respect to x.

First, let's find the value of f(a) at a = 3.14159265359:

f(a) = 7cos(a)
f(3.14159265359) = 7cos(3.14159265359) ≈ -7

Next, let's find the value of f'(a) at a = 3.14159265359:

f'(x) = -7sin(x)
f'(a) = -7sin(a)
f'(3.14159265359) = -7sin(3.14159265359) ≈ 0

Now we have all the pieces we need to plug into the formula for L(x):

L(x) = f(a) + f'(a)(x - a)
L(x) = -7 + 0(x - 3.14159265359)
L(x) = -7

So the linearization of the function f(x) = 7cos(x) at a = 3.14159265359 is:

L(x) = -7

To find the linearization L(x) of the function f(x) = 7cos(x) at a specific point a, we'll use the formula:

L(x) = f(a) + f'(a)(x - a)

Given that a = 3.14159265359 (approximating π), first we need to find f(a) and f'(a).

1. f(a) = 7cos(a) = 7cos(3.14159265359) ≈ -7
2. To find f'(x), we take the derivative of f(x):
f'(x) = -7sin(x)

Now, we can find f'(a):
f'(a) = -7sin(3.14159265359) ≈ 0

Finally, we can plug these values into the linearization formula:
L(x) = -7 + 0(x - 3.14159265359)

Simplifying, we get:

L(x) = -7

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(a) The expression for the total area of the tablet = (8 + 2z)(5 + 2z)

(b) Equation is: (8 + 2z)(5 + 2z) = 50.3125 and the solution to this equation refers to the thickness of frame for which the area of the tablet is 50.3125.

(c) Solution or the thickness of the frame must be 0.375 inches.

The dimensions of the screen of a tablets are 8 inches by 5 inches.

border around the screen has thickness z.

So the length with frame = 8 + 2z

and the width of the screen with frame = 5 + 2z

So the expression for the total area of the tablet = Length* Width = (8 + 2z)(5 + 2z)

Equation for which the expression is equal to 50.3125 is given by,

(8 + 2z)(5 + 2z) = 50.3125

So the solution to this equation refers to the thickness of frame for which the area of the tablet is 50.3125.

Solving the equation we get,

(8 + 2z)(5 + 2z) = 50.3125

40 + 10z + 16z + 4z² = 50.3125

4z² + 26z - 10.3125 = 0

Solving this quadratic equation we get the solutions,

z = -6.875, 0.375

Since the thickness cannot be negative so -6.875 must be neglected.

Hence the thickness of the frame is 0.375 inches.

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The arc length of the polar curve r = 8θ^2 on the interval [0, 2] is approximately 70.71 units.

The polar curve r = 8θ^2 on the interval [0, 2] has an arc length which can be calculated using the formula for arc length in polar coordinates:

L = ∫√(r^2 + (dr/dθ)^2) dθ, from θ = 0 to θ = 2.
First, we need to find the derivative dr/dθ:

r = 8θ^2, so dr/dθ = 16θ.

Now, plug r and dr/dθ into the arc length formula:
L = ∫√((8θ^2)^2 + (16θ)^2) dθ, from θ = 0 to θ = 2.

Simplify the integrand:

L = ∫√(64θ^4 + 256θ^2) dθ, from θ = 0 to θ = 2.

Factor out 64θ^2:

L = ∫√(64θ^2(1 + θ^2)) dθ, from θ = 0 to θ = 2.

Now, apply the substitution u = 1 + θ^2, so du = 2θ dθ:

L = 32∫√(u) du, from u = 1 to u = 5.

Integrate and evaluate:

L = (32/3)(u^(3/2)) | from u = 1 to u = 5.

L = (32/3)(5^(3/2) - 1^(3/2)).

L ≈ 70.71 units.

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The critical points and their classifications are: (0, kπ/2), local minimum for all k.

To find the critical points of f(x, y), we need to find where the partial derivatives of f with respect to x and y are equal to zero:

∂f/∂x = 6x = 0

∂f/∂y = 2sin(2y) = 0

From the first equation, we get x = 0, and from the second equation, we get sin(2y) = 0, which has solutions y = kπ/2 for any integer k.

So the critical points are (0, kπ/2) for all integers k.

To classify these critical points, we need to use the second derivative test. The Hessian matrix of f is:

H = [6 0]

[0 -4sin(2y)]

At the critical point (0, kπ/2), the Hessian becomes:

H = [6 0]

[0 0]

The determinant of the Hessian is 0, so we can't use the second derivative test to classify the critical points. Instead, we need to look at the behavior of f in the neighborhood of each critical point.

For any k, we have:

f(0, kπ/2) = 1 + 3(0)² - cos(2kπ) = 2

So all the critical points have the same function value of 2.

To see whether each critical point is a maximum, minimum, or saddle point, we can look at the behavior of f along two perpendicular lines passing through each critical point.

Along the x-axis, we have y = kπ/2, so:

f(x, kπ/2) = 1 + 3x² - cos(2kπ) = 1 + 3x²

This is a parabola opening upwards, so each critical point (0, kπ/2) is a local minimum.

Along the y-axis, we have x = 0, so:

f(0, y) = 1 + 3(0)² - cos(2y) = 2 - cos(2y)

This is a periodic function with period π, and it oscillates between 1 and 3. So for each k, the critical point (0, kπ/2) is neither a maximum nor a minimum, but a saddle point.

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(a) The marginal cost function is C'(x) = 5 + 0.16x + 0.0012x².

(b) C'(100) = 21.00. This predicts the approximate cost of the 101st pair of jeans.

(c) The difference between C'(100) and the actual cost of manufacturing the 101st pair of jeans is $20.51.

(a) To find the marginal cost function, we need to take the derivative of the cost function C(x) with respect to x:

C'(x) = 5 + 0.16x + 0.0012x²

(b) To find C'(100), we substitute x = 100 into the marginal cost function:

C'(100) = 5 + 0.16(100) + 0.0012(100)²
C'(100) = 21.00

This predicts the approximate cost of the 101st pair of jeans, as the marginal cost function tells us the cost of producing one additional unit (pair of jeans) at a given quantity. So, we can use C'(100) to estimate the cost of producing the 101st pair of jeans.

To find the approximate cost of the 100th pair of jeans, we can substitute x = 100 into the cost function C(x):

C(100) = -1700 + 5(100) + 0.08(100)² + 0.0004(100)³
C(100) = $2330

So, the approximate cost of the 100th pair of jeans is $2330.

To find the approximate cost of the 101st pair of jeans, we can add C'(100) to the cost of producing 100 pairs of jeans:

C(100) + C'(100) = $2330 + $21.00
C(101) ≈ $2351

So, the approximate cost of the 101st pair of jeans is $2351.

To find the exact cost of the 99th pair of jeans, we can substitute x = 99 into the cost function C(x):

C(99) = -1700 + 5(99) + 0.08(99)² + 0.0004(99)³
C(99) = $2288.44

So, the exact cost of the 99th pair of jeans is $2288.44.

(c) To find the difference between C'(100) and the actual cost of manufacturing the 101st pair of jeans, we need to subtract the cost of producing 100 pairs of jeans from the cost of producing 101 pairs of jeans:

C(101) - C(100) = [-1700 + 5(101) + 0.08(101)² + 0.0004(101)³] - [-1700 + 5(100) + 0.08(100)² + 0.0004(100)³]
C(101) - C(100) = $20.51

So, the difference between C'(100) and the actual cost of manufacturing the 101st pair of jeans is $20.51.

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a) Two different twin circles for five pair of twins are ( 5,2,4,2,3,5,4,3,1,1,3) and ( 3,1,1,3,4,5,3,2,4,2,5).

b) No twin circles in 3 pair of twins because any of arrangement of them cannot fulfil the condition of twin circle.

c) The partial twin circle ( third circle) present in above figure can't be completed because 4 positions are fixed there and after that number of persons more than seats.

We have a pair twins are numbered 1, 1, 2, 2, 3, 3, and so on. They all seated in a circle so that the least number of gaps between two twins always

equals their assigned number. This is called a twin circle. That is Number of persons between 1 and 1 twins = 0

Number of persons between 2 and 2 twins = 1,

so on.. Reflections (flips) and rotations (turns) of a twin circle are regarded as the same.

a) We have to make two twin circles for five pair twins. The arrangement of pair twins in two different ways with the satisfaction of conditions. So, first arrangement is ( 5,2,4,2,3,5,4,3,1,1,3) and

other arrangement is ( 3,1,1,3,4,5,3,2,4,2,5).

b) There is no twin circle between the arrangement of 3 twin pairs. Because in case of 3 twin pair total members = 3×2 = 6 and number of members can be seat between pairs are 3( 1+2+0). As we know, it is fixed that no person between (1,1). So, we cannot be arrange the 2 pairs with desirable 3 gaps that is 1 person between (2,2) and 2 persons between (3,3).

c) There is total 12 positions to seat in circles. The position of 6 and 1 is fixed. According to above scenario, position next to 1 is for 1 (clockwise) and 5th position from given 1 position in (clockwise) is other member of twin 6. Now, four positions are fixed. Eight positions are left and 4 twin pairs (2,2) , (3,3), (4,4),(5,5). Number of persons seat between 4 pairs are 10 in counts ( greater than position ) so, no such arrangement is possible. Hence, this partial circle can't be completed.

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Complete question:

The above figure complete question.

Pairs of twins are numbered 1, 1, 2, 2, 3, 3, and so on. They are seated in a circle so that the least number of gaps between two twins always

equals their assigned number. This is called a twin circle. Note that this means there is no person between the twins numbered 1 and 1, there is just one person between the twins numbered 2 and 2, and so on. Reflections (flips) and rotations (turns) of a twin circle are regarded as the same. For example, the following are the same twin circles for 4 pairs of twin.

a) Find two twin circles for five pairs of twin

b) Explain why no twin circles in 3 pairs of twin

c) explain why this partial twin circle can't be completed ? ( third circle)

The sum of all the angles in a quadrilateral is 360° ,So The angle measure indicated is 137°.

What is radius?

In classical geometry, the radius of a circle or sphere is any line segment that links the object's centre to its edge; in more modern usage, the term also refers to the length of such line segments. The Latin term "radius," which may also be used to describe a chariot wheel spoke, is where the word "radius" first appeared.

The length of tangents drawn from an external point is known to be constant. The circle's radius across the point of contact and any other point on the circle are perpendicular to the tangent. A quadrilateral has 360° of angles total. In light of this, 137° is the indicated angle measurement.

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Correct question is

Find the angle measure indicated. Assume that lines which appear to be tangent are tangent.

The diagram to create a similar triangle has been attached.

Similar triangles are defined as triangles that possess the same shape, but then their sizes will likely vary. We can also say that two triangles are referred to as similar if they possess the same ratio of its' corresponding sides and also an equal pair of corresponding angles

The two different triangles can be formed by placing the 90° angle adjacent to, or opposite the given side.

In the diagram below attached, we see that the two triangles are ABC and ABD. Thus, the right angles are located at vertex C and vertex B, respectively.

Thus, it has been created with the given measurements

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The expression is (5/2)b + 7 for muffins is made by the baking company  in total in one night.

To find the total number of muffins the baking company made in one night, we can use the following expression:

Total = b - (b/2) + 3b + 7

Let's break it down by each hour:

- In the first hour, the company made b muffins.
- In the second hour, they threw away half of the muffins made in the first hour, which is b/2. So, they only have b - (b/2) muffins left.
- In the third hour, they made 3 times as much as the first two hours, which is 3b.
- In the last hour, they made 7 more muffins.

If we simplify the expression by combining like terms, we get:

Total = (5/2)b + 7

Therefore, the baking company made (5/2)b + 7 muffins in total in one night.

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The car mechanic is correct in saying that they will have used over 80% of the oil in the tin after 9 weeks.

To determine if the car mechanic is correct, we first need to calculate how much oil they will use in 9 weeks.

450 millilitres of oil are used each week, so after 9 weeks, they will have used:

450 x 9 = 4050 millilitres

Next, we need to convert this to litres, since the oil tin is measured in litres.

There are 1000 millilitres in 1 litre, so:

4050 ÷ 1000 = 4.05 litres

Therefore, after 9 weeks, the car mechanic will have used 4.05 litres of oil.

Now we need to determine if this is over 80% of the total oil in the tin.

The tin contains 5 litres of oil, so we need to find 80% of 5:

5 x 0.8 = 4

So if the car mechanic has used more than 4 litres of oil in 9 weeks, they have used over 80% of the oil in the tin.

We know from earlier that they will have used 4.05 litres, which is slightly over 80%. Therefore, the car mechanic is correct in saying that they will have used over 80% of the oil in the tin after 9 weeks.

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Answer:

Step-by-step explanation:

Of means to multiply

So to find .3 of the 10.38 pounds up apples:

.3 x 10.38

=3.114 pounds of apples were used

The number of possible outcomes of the compound event of selecting a card, spinning the spinner, and tossing a coin is B. 72 outcomes.

To determine the number of possible outcomes for the compound event, we need to multiply the number of outcomes for each individual event.

There are 12 cards labeled 1 through 12, so there are 12 possible outcomes for selecting a card. The spinner is divided into three equal-sized portions, so there are 3 possible outcomes for spinning the spinner. There are 2 possible outcomes for tossing a coin (heads or tails).

the total number of possible outcomes for the compound event:

12 (selecting a card) x 3 (spinning the spinner) x 2 (tossing a coin) = 72

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First, solve the expression inside the parentheses: (-28) - (+42) = -70

Next, subtract 3 from -70: -70 - (+3) = -73

The given expression can be simplified as follows:

[( -28 )-( +42)]-(+3) = -28 - 42 - 3 = -73

So, the answer to the given expression is -73.

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The logarithmic expression log7 18 is equivalent to log 18 / log 7 using the change-base formula.

The change-base formula states that the logarithm of a number to a certain base can be converted to the logarithm of the same number to a different base by dividing the logarithm of the number to the first base by the logarithm of the number to the second base.

In this case, we want to convert log7 18 to a logarithm with base 10. Therefore, using the change-base formula, we can write:

log7 18 = log 18 / log 7

Using a calculator, we can evaluate the right-hand side of the equation to get:

log7 18 = 1.2553 / 0.8451

log7 18 = 1.4845 (rounded to four decimal places)

Therefore, the logarithmic expression log7 18 is equivalent to log 18 / log 7, which is approximately equal to 1.4845.

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The distance between the two cruise liners is approximately 3.6 km.

We can use the Pythagorean theorem to find the distances between Liam and the two cruise liners, and then use the distance formula to find the distance between the two cruise liners. Let's call the distance between Liam and the first cruise liner "d1" and the distance between Liam and the second cruise liner "d2". Then:

d1 = sqrt(5² - 2²) = sqrt(21) km

d2 = sqrt(6.8² - 2²) = sqrt(44.44) km

To find the distance between the two cruise liners, we can use the distance formula:

distance = sqrt((d2 - d1)² + (6.8 - 5)²) km

Plugging in the values, we get:

distance = sqrt((sqrt(44.44) - sqrt(21))² + 1.8²) km

Simplifying this expression gives:

distance = sqrt(44.44) - sqrt(21) km

So the distance between the two cruise liners is approximately 3.9 km.

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The height of the triangle is given as follows:

[tex]h = 2\sqrt{3}[/tex] cm.

The area of the triangle is given as follows:

[tex]A = 4\sqrt{3}[/tex] cm².

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

The theorem is expressed as follows:

c² = a² + b².

In which:

Considering an equilateral triangle, in which all the side lengths are of 4, we have a right triangle in which:

Hence the height is obtained as follows:

h² + 2² = 4²

h² = 12

[tex]h = \sqrt{3 \times 4}[/tex]

[tex]h = 2\sqrt{3}[/tex] cm

The area of a triangle is given as half the multiplication of the base and of the height, hence:

[tex]A = 0.5 \times 4 \times 2 \sqrt{3}[/tex]

[tex]A = 4\sqrt{3}[/tex] cm².

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To calculate the value of Kearney's retirement savings when he retired, we need to use the formula for compound interest:

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

where:
A = final amount
P = initial principal (the amount Kearney invested each month)
r = annual interest rate (5.5%)
n = number of times interest is compounded per year (12, since we're assuming monthly compounding)
t = number of years

First, we need to calculate the total number of payments Kearney made into his retirement savings:

68 - 30 = 38 years

Since Kearney made monthly payments, the total number of payments is:

38 years x 12 months/year = 456 payments

Next, we need to calculate the value of each payment after it has earned interest. We can use the same formula as above, but with t = 1 (since we're calculating the value of one payment period):

P' = P(1 + r/n)^(nt)
P' = 200(1 + 0.055/12)^(12*1)
P' = 200(1.00458333333)^12
P' = 200(1.00458333333)^12
P' = 200(1.00458333333)^12
P' = 243.382740047

So each $200 payment is worth $243.38 after one month of earning interest.

Now we can use the formula for the future value of an annuity to calculate the total value of Kearney's retirement savings:

A = P'[(1 + r/n)^(nt) - 1]/(r/n)
A = 243.38[(1 + 0.055/12)^(12*38) - 1]/(0.055/12)
A = 243.38[1.93378208462 - 1]/(0.055/12)
A = 243.38[34.3478377249]
A = $8,351.53

Therefore, the value of Kearney's retirement savings when he retired was approximately $8,351.53.

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When Kearney retired at age 68, the value of his retirement savings was $557,123.35.

To find the value of Kearney's retirement savings when he retired, we'll use the Future Value of an Annuity formula. Here are the given values and the formula:

Monthly investment (PMT) = $200

Annual interest rate (r) = 5.5% = 0.055

Monthly interest rate (i) = (1 + r)^(1/12) - 1 ≈ 0.004434

Number of years of investment (n) = 68 - 30 = 38 years

Number of months of investment (t) = 38 years * 12 months = 456 months

Future Value of Annuity (FV) formula:

FV = PMT * [(1 + i)^t - 1] / i

Now, we'll plug in the values and calculate the Future Value:

FV = 200 * [(1 + 0.004434)^456 - 1] / 0.004434

FV ≈ 200 * [12.2883] / 0.004434

FV ≈ 557123.35

The value of his retirement savings was approximately $557,123.35.

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Answer:

Step-by-step explanation:

To find the solution set to the inequality w - 10 < 16, we can solve for w by adding 10 to both sides of the inequality:

w - 10 + 10 < 16 + 10 w < 26

This means that any number less than 26 is part of the solution set to the inequality. So, out of the given options, the number that is not part of the solution set is D. 27 because it is greater than 26.

For a purchase of a new desktop computer for $1250, loan description that would result in the smallest amount of interest she would have to pay is 12 months at 6. 25% annual simple interest rate. Therefore, the correct option is option 1.

To determine which loan description results in the smallest amount of interest for Roxie, we'll calculate the interest for each option using the simple interest formula:

Interest = Principal × Rate × Time.

1. 12 months at 6.25% annual simple interest rate:

Interest = $1250 × 6.25% × (12/12)

Interest = $1250 × 0.0625 × 1

Interest = $78.13

2. 18 months at 6.75% annual simple interest rate:

Interest = $1250 × 6.75% × (18/12)

Interest = $1250 × 0.0675 × 1.5

Interest = $126.56

3. 24 months at 6.5% annual simple interest rate:

Interest = $1250 × 6.5% × (24/12)

Interest = $1250 × 0.065 × 2

Interest = $162.50

4. 30 months at 6.00% annual simple interest rate:

Interest = $1250 × 6.00% × (30/12)

Interest = $1250 × 0.06 × 2.5

Interest = $187.50

Comparing the interest amounts, the smallest interest is for the first option, 12 months at 6.25% annual simple interest rate, with an interest amount of $78.13.

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The z-score of the customer that just turned 25 years old is  -1.45. The z-score for an age of 75 years is approximately 2.17, which is greater than 2,  Since a z-score greater than 2 represents a considerable deviation.

(a)

To find the z-score for a customer that just turned 25 years old :

z-score = (x - mean) / standard deviation

Plugging in the values, we get:
z-score = (25 - 45) / 13.8 = -1.45, where x = 25 years, mean  = 45 years, and standard deviation = 13.8 years.
Rounding to the nearest hundredth, the z-score is -1.45.

(b)

To find an example of a customer age with a z-score greater than 2, we need to identify an age that deviates significantly from the mean given the standard deviation. Since a z-score greater than 2 represents a considerable deviation, let's consider an age of 75 years.

Using the same formula as before:

z = (x - μ) / σ

where:

   x is the customer's age (75 years),

   μ is the mean of the distribution (45 years),

   σ is the standard deviation of the distribution (13.8 years).

Calculating the z-score:

z = (75 - 45) / 13.8

z = 2.17

The z-score for an age of 75 years is approximately 2.17, which is greater than 2, fulfilling the requirement of the question.

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The venue breaks even when 1,250 people buy tickets, and the total value of tickets sold at that point is $100,000.

To find the break-even point for the venue, we need to set the musician's charges (C(x) = 64x + 20,000) equal to the venue's earnings from ticket sales ($80 per ticket).  Hence,

1. Set the musician's charges equal to the venue's earnings:

64x + 20,000 = 80x

2. Subtract 64x from both sides:

20,000 = 16x

3. Divide both sides by 16:

x = 1,250

At the break-even point, 1,250 people need to buy tickets. To find the value of the total tickets sold at this point:

1. Multiply the number of attendees (x) by the ticket price:

Total ticket sales = x * ticket price

2. Substitute the values:

Total ticket sales = 1,250 * $80

3. Calculate the total ticket sales:

Total ticket sales = $100,000

So, the breaks even point is 1,250 people buying tickets, and corresponding total value of tickets sold is $100,000.

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The p-value for the hypothesis test is approximately 0.000006.

To find the p-value, we follow these steps:

1. State the null hypothesis (H0) and alternative hypothesis (H1):
  H0: p = 0.5
  H1: p > 0.5

2. Calculate the sample proportion (p-hat): p-hat = 120/200 = 0.6

3. Calculate the test statistic (z) using the formula: z = (p-hat - p) / √((p * (1 - p)) / n)
  z = (0.6 - 0.5) / √((0.5 * 0.5) / 200) ≈ 2.683

4. Find the corresponding p-value using a z-table or calculator. The area to the right of the test statistic (2.683) is approximately 0.000006.

Since the p-value (0.000006) is less than the significance level (typically 0.05), we reject the null hypothesis, indicating that the true proportion of apple weights larger than 100 grams is larger than 0.5.

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Therefore, the estimate is quite close to the exact value, with an error of about 0.000450.

We can use differentials to estimate the value of ⁴√1.3 as follows:

Let y = ⁴√x, then we have:

dy/dx = 1/(4x^(3/4))

We want to estimate the value of y when x = 1.3, so we have:

Δy ≈ dy * Δx

where Δx = 0.3 - 1 = -0.7 (since we are approximating 1.3 as 1)

Substituting the values, we get:

Δy ≈ (1/(4(1)^3/4)) * (-0.7) ≈ -0.219

Hence, the estimate for ⁴√1.3 is:

y ≈ ⁴√1 + Δy ≈ 0.780

The exact value of ⁴√1.3 is approximately 0.780450255.

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The solutions to the triangle PQR are:

Angle Q ≈ 112°

Side QR ≈ 8.98

Side PR ≈ 13.71

To solve the triangle PQR, we can use the fact that the sum of the angles in a triangle is always 180°. So we can find angle Q by subtracting the measures of angles P and R from 180°:

angle Q = 180° - angle P - angle R

angle Q = 180° - 42° - 26°

angle Q = 112°

Now, we can use the law of sines to find the lengths of the sides QR and PR.

The law of sines states that in any triangle ABC, the following equation holds:

a/sin(A) = b/sin(B) = c/sin(C)

where a, b, and c are the side lengths of the triangle, and A, B, and C are the opposite angles, respectively.

Applying this formula to triangle PQR, we can write:

QR/sin(R) = PQ/sin(Q)

QR/sin(26°) = 19/sin(112°)

Solving for QR, we get:

QR = (19 × sin(26°))/sin(112°)

QR ≈ 8.98

Similarly, we can find PR by applying the law of sines to triangle PQR as follows:

PR/sin(P) = PQ/sin(Q)

PR/sin(42°) = 19/sin(112°)

Solving for PR, we get:

PR = (19 × sin(42°))/sin(112°)

PR ≈ 13.71

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