This graph represents the equation y=(x-5)^2-1 .




How many ordered pairs (x, y) for 3 < x < 7 satisfy this equation?

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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The exact value of sin⁻¹(−1/2) is -π/6.

Given, sin⁻¹(-1/2)

The inverse sine function, sin⁻¹, or arcsin, returns the angle whose sine is equal to the given value. In this case, we are looking for the angle whose sine is -1/2.

Let y = sin⁻¹(-1/2)

sin (y) = -1/2

sin (y) = - sin (π/6)

sin (y) =  sin (- π/6)

y = - π/6

sin⁻¹(-1/2) = - π/6

To understand why the answer is -π/6, we can consider the unit circle. On the unit circle, the sine function represents the y-coordinate of a point corresponding to an angle. For -1/2, we need to find the angle where the y-coordinate is -1/2.

One such angle is -π/6, where the point on the unit circle is located in the fourth quadrant. At this angle, the y-coordinate is -1/2. Hence, sin⁻¹(−1/2) is -π/6.

Therefore, the exact value of sin⁻¹(−1/2) is -π/6.

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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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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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Answer:

C

Step-by-step explanation:

The lowest point on the graph on the y-axis is -4

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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(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 equation of the line representing that best fit the given scatterplot is given by option d. y = -0.005x + 22.5.

Consider the two points from the attached scatterplot.

Let the coordinates of the two point be ( x₁ , y₁) = ( 1500 , 12.5 )

And other point be ( x₂ , y₂) = ( 2000 , 10 )

Slope of the line 'm'  = ( y₂ - y₁ ) / ( x₂ - x₁ )

                                  = ( 10 - 12.5) / ( 2000 - 1500 )

                                  = -2.5 / 500

                                  = -0.005

From the attached scatterplot we have,

y-intercept 'c' where x = 0 is equals to 22.5.

The equation best which represents the line of best fit for the scatterplot is equals to,

y = mx + c

Substitute the value we have,

y = -0.005x + 22.5

Therefore, the equation of the line representing scatterplot is equals to option d. y = -0.005x + 22.5.

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The above question is incomplete, the complete question is:

Which equation best represents the line of best fit for the scatterplot?

a) y = 2.5x + 25 b) y = −2.5x + 20 c) y = −0.05x + 20 d) y = −0.005x + 22.5

Attached scatterplot.

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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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 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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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.

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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The length of the minor arc GD is approximately 0.196π units.

To get the length of the minor arc GD, we need to subtract the measure of the major arc mDUG from the circumference of the circle, and then divide by 360° to find the length of one degree of arc.
First, we need to find the circumference of the circle. Since GA = 29, we know that the radius of the circle is also 29. The formula for the circumference of a circle is C = 2πr, so for this circle we have: C = 2π(29) = 58π
Next, we need to subtract the measure of the major arc mDUG from the circumference of the circle. Since mDUG = 185°, we have:
58π - (185/360)(58π) = (175/360)(58π)
Simplifying this expression, we get: (175/360)(58π) = 29(175/72)π ≈ 70.48π
Finally, we divide this value by 360° to find the length of one degree of arc:
(70.48π)/360 ≈ 0.196π
Therefore, the length of the minor arc GD is approximately 0.196π units.

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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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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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3) The expression In(4x^5) - In(x^3) - In 4 can be simplified using the properties of logarithms. We know that ln(a) - ln(b) = ln(a/b) and ln(a^n) = n ln(a), so we can write:In(4x^5) - In(x^3) - In 4 = In[(4x^5)/(x^3)] - In 4= In(4x^2) - In 4= In(4x^2/4)= In(x^2)Thus, the simplified expression is In(x^2).4) To solve this problem, we need to use the formula for compound interest:A = P(1 + r/n)^(nt)where A is the final amount, P is the initial amount, r is the interest rate (as a decimal), n is the number of times interest is compounded per year, and t is the number of years.We want to find t when A = 3P and r = 0.02 (since the price of beef has inflated by 2%). We are told that interest is compounded biannually, so n = 2. Plugging in these values and solving for t, we get:3P = P(1 + 0.02/2)^(2t)3 = (1.01)^2tln(3) = ln(1.01^2t)ln(3) = 2t ln(1.01)t = ln(3) / (2 ln(1.01))Using a calculator, we find t ≈ 34.64 years. Therefore, it will take about 34.64 years for the price of beef to triple at a 2% biannual inflation rate.

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It will take approximately 110 years for the price of beef to triple when inflating 2% compounded biannually.

3) To simplify the expression In(4x^5) - In(x^3) - In(4), we will use the properties of logarithms:

- In(a) - In(b) = In(a/b)
- In(a^b) = b * In(a)

So, we can rewrite the expression as:

In(4x^5 / (x^3 * 4))

Now, we can simplify the expression inside the natural logarithm:

(4x^5) / (4x^3) = x^(5-3) = x^2

Thus, the simplified expression is:

In(x^2)

4) To find how long it will take for the price of beef to triple when inflating 2% compounded biannually, we can use the compound interest formula:

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

where A is the final amount, P is the initial amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years. In this case, we want the final amount to be triple the initial amount:

3P = P(1 + 0.02/2)^(2t)

To solve for t, we can divide both sides by P:

3 = (1 + 0.01)^(2t)

Now, take the natural logarithm of both sides and use the properties of logarithms:

ln(3) = ln((1 + 0.01)^(2t))
ln(3) = 2t * ln(1 + 0.01)

Finally, isolate t:

t = ln(3) / (2 * ln(1 + 0.01))

t ≈ 109.96

It will take approximately 110 years for the price of beef to triple when inflating 2% compounded biannually.

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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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If Kai spent 55$ all together then each pack of gum cost $4.

To solve this question follow the steps given below:

Calculate the total cost of Gatorade.
Since there are 5 bags and each bag has a bottle of Gatorade that costs $3, the total cost for Gatorade is 5 * $3 = $15.

Calculate the total cost of gum.
Since Kai spent $55 in total, we need to subtract the cost of Gatorade to find the total cost of gum. $55 - $15 = $40.

Calculate the total number of gum packs.
Each bag contains 2 packs of gum, and there are 5 bags. So, there are    2 * 5 = 10 packs of gum.

Calculate the cost of each pack of gum.
To find the cost of each pack of gum, divide the total cost of gum by the number of gum packs. $40 / 10 = $4.

So, each pack of gum cost $4.

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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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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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The integral  ∫(8x/2 + 2/x^3)dx evaluates to 4x^2 - 2/x^2 + C, where C is the constant of integration.

The given integral ∫(8x/2 + 2/x^3)dx is definite integral without any integration limits. To evaluate this integral, we can split it into two parts

∫8x/2 dx + ∫2/x^3 dx

We made use of the power rule of integration to simplify the first term, and the inverse power rule to simplify the second term.

Simplifying each integral, we get

4x^2 - 2/x^2 + C

where C is the constant of integration.

Therefore, the final answer to the integral is

∫(8x/2 + 2/x^3)dx = 4x^2 - 2/x^2 + C

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--The given question is incomplete, the complete question is given

" Evaluate. Assume that x>0. ∫(8x/2 + 2/x^3)dx"--

Answer:

Step-by-step explanation:

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.

A BASE jumper will fall 324 feet in 4.5 seconds.

What are velocity ?

velocity is a unit of measurement for the Distance an object travels in a

the predetermined period of time. Here is a word equation that illustrates the connection between space, speed, and time: velocity is calculated by dividing the total Distance traveled by the journey time.

We can use the given function to find out how far a BASE jumper will fall in 4.5 seconds:

d = 16t²

where d is the distance (in feet) and t is the time (in seconds).

Substitute t = 4.5 into the formula:

d = 16(4.5)²

d = 324

Therefore, a BASE jumper will fall 324 feet in 4.5 seconds.

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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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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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