Evaluate h'(9) where h(x) = f(x) · g(x) given the following.• f(9) = 9• f '(9) = −1.5• g(9) = 3• g'(9) = 2h'(x) =

Cos 165 in terms of sine and cosine of acute angle would give cos(165) = -(1 + √3) / (2√2).

To find the cosine of 165 degrees in terms of sine and cosine of an acute angle, we can use the cosine angle addition formula:

cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

Since 120 degrees is in the second quadrant, the cosine is negative, and the sine is positive:

cos(120) = -cos(60) and sin(120) = sin(60)

cos(165) = -cos(60)cos(45) - sin(60)sin(45)

Now we can plug in the values of the trigonometric functions:

cos(165) = - (1/2) x (1/√2) - (√3/2) x (1/√2)

cos(165) = -(1 + √3) / (2√2)

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The difference between the means expressed as a multiple of the mean absolute deviation is 2.3.

To find the difference between the means expressed as a multiple of the mean absolute deviation, we need to calculate the absolute difference between the two means and divide it by the mean absolute deviation.

The absolute difference between the means is:

|30.2 - 25.6| = 4.6

To express this difference as a multiple of the mean absolute deviation, we divide it by the mean absolute deviation:

4.6 / 2 = 2.3

Therefore, the difference between the means expressed as a multiple of the mean absolute deviation is 2.3.

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The ladder reaches up 20ft the side of the house.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

This theorem can be used to solve problems involving right triangles, such as finding the length of the sides or the height of an object.

In this problem, we are given the length of the ladder and the height up the side of the house that it reaches.

We can use the Pythagorean theorem to find the distance from the base of the ladder to the side of the house.

We can then use this distance and the height up the side of the house that the ladder reaches to find the length of the ladder using the Pythagorean theorem again.

Let's call the distance from the base of the ladder to the side of the house "x". We can then use the Pythagorean theorem to find the height that the ladder reaches up the side of the house.

According to the Pythagorean theorem, the length of the ladder (which is the hypotenuse of the right triangle formed by the ladder, the ground, and the side of the house) is equal to the square root of the sum of the squares of the other two sides.

So, if we let "h" be the height up the side of the house that the ladder reaches, we have:

ladder length = √(x^2 + h^2)

We know that the ladder is 20ft long and reaches up 16ft, so we can set up the equation:

20 = √(x^2 + 16^2)

Squaring both sides of the equation, we get:

400 = x^2 + 256

Subtracting 256 from both sides, we get:

144 = x^2

Taking the square root of both sides, we get:

x = 12

So the ladder is leaning against the house 12ft away from the base, and we can use the Pythagorean theorem to find the height up the side of the house that the ladder reaches:

ladder length = √(12^2 + 16^2) = √(144 + 256) = √400 = 20

Therefore, the ladder reaches up 20ft the side of the house.

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Answer:

no it is not possible

Step-by-step explanation:

To do this, we can use the Pythagorean theorem and trigonometric ratios instead.

1. Determine the coordinates of the given point, let's call it P(x, y), and the center of the circle, let's call it O(h, k). Also, note the radius, r.

2. Calculate the distance between point P and the center O using the Pythagorean theorem: d^2 = (x-h)^2 + (y-k)^2, where d is the distance.

3. Set d equal to the radius of the circle: r^2 = (x-h)^2 + (y-k)^2.

4. Now, let's find the angle θ between the x-axis and the line OP without using arctan. To do this, we'll use the sine and cosine ratios:

sin(θ) = (y-k) / r and cos(θ) = (x-h) / r

5. To eliminate the need for arctan, we can use the Pythagorean identity sin^2(θ) + cos^2(θ) = 1. Substitute the sine and cosine ratios we found earlier:

((y-k) / r)^2 + ((x-h) / r)^2 = 1

6. Simplify the equation by multiplying both sides by r^2:

(y-k)^2 + (x-h)^2 = r^2

You'll notice that this equation is the same as the one we found in step 3, confirming that the point P lies on the circle. You've now solved the point circle problem without using arctan, by employing the Pythagorean theorem and trigonometric ratios instead.

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Answer:

Step-by-step explanation:

In option (a), sin(t) < 0 and cos(t) < 0, In trigonometry, the terminal point of an angle t is the point on the unit circle where the angle intersects with the circle.

The position of the terminal point determines the quadrant in which the angle lies.

To determine the quadrant, we need to look at the signs of the sine and cosine functions. In quadrant I, both sine and cosine are positive. In quadrant II, sine is positive and cosine is negative. In quadrant III, both sine and cosine are negative. In quadrant IV, sine is negative and cosine is positive.

In option (a), sin(t) < 0 and cos(t) < 0, both the sine and cosine functions are negative. This means that the terminal point lies in quadrant III.

In option (b), sin(t) > 0 and cos(t) < 0, the sine function is positive and the cosine function is negative. This means that the terminal point lies in quadrant II.

In option (c), sin(t) > 0 and cos(t) > 0, both the sine and cosine functions are positive. This means that the terminal point lies in quadrant I.

In option (d), sin(t) < 0 and cos(t) > 0, the sine function is negative and the cosine function is positive. This means that the terminal point lies in quadrant IV.

In summary, the signs of the sine and cosine functions can be used to determine the quadrant in which the terminal point lies.

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If he drives at an average speed of 70 km/h for 45 min, the distance of the journey from Mr. Linden's home to his office is 52.5 km.

To find the distance of Mr. Linden's journey from his home to his office, we can use the formula:

Distance = Speed x Time

Since Mr. Linden drives at an average speed of 70 km/h for 45 minutes, we first need to convert the time to hours:

45 minutes = 0.75 hours

Now, we can plug in the values we have into the formula:

Distance = 70 km/h x 0.75 hours
Distance = 52.5 km

Therefore, the distance of the journey from Mr. Linden's home to his office is 52.5 km.

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1. An equation representing the number (N) of novels Deshaun needs to read in M months is N = 3M.

2. Based on the above equation, Deshaun needs to read 57 novels in 19 months.

An equation is a mathematical statement that shows the equality or equivalence of mathematical expressions.

While mathematical expressions combine variables with numbers, constants, and values using mathematical operands, equations use the equal symbol (=) in addition.

The number of novels Deshaun needs to read per month = 3

The number of months involved = 19 months

Let the number of novels Deshaun needs to read in M months = N

Let the number of months involved = M

N = 3M

N = 57 (3 x 19)

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y=(x+3)^2-1

have a good day :)

Answer: F(x) = (x + 3)^2 - 1

Step-by-step explanation:

The value of the limit lim x->2 | (x-2)(x+2)/(x^2-4) is undefined. This is because as x approaches 2, the denominator (x^2-4) approaches 0, which means that the fraction as a whole is undefined. Therefore, there is no value that the limit can approach.

The value of the limit lim x->2 | (x-2)(x+2)/(x^2-4) is:
Step 1: Recognize that the given expression can be simplified. Notice that the denominator, x^2 - 4, is a difference of squares, so it can be factored as (x-2)(x+2).
Step 2: Simplify the expression by canceling the common factors in the numerator and the denominator: (x-2)(x+2) / (x-2)(x+2) simplifies to 1, because the factors (x-2)(x+2) cancel each other out.
Step 3: Now that the expression is simplified, substitute x = 2 to find the value of the limit: lim x->2 | 1 = 1.
Your answer: The value of the limit lim x->2 | (x-2)(x+2)/(x^2-4) is 1.

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The coordinates of the point prior to the dilation are (-2,-1) when the Scale factor is 1/3 and the new coordinates are (-6,3).

To find the coordinates of the point prior to the dilation, we need to use the formula for dilation:

(x’, y’) = (k x, ky)

where

(x’, y’) = the new coordinates

(x, y) = original coordinates

k = scale factor

Given data:

Scale factor =  1/3

New coordinates = (-6, 3)

By substuting the values in the equation we get:

(-6, 3) = (k x, ky)

Solving for x and y:

k x = -6

ky = 3

Dividing the ky equation by the k x  equation we get:

y/x = 3/-6

y/x = -1/2

From the above equation, we can assume that x = 2 and y = -1.

Therefore, the coordinates of the point prior to the dilation are (-2,-1).

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Answer:

Step-by-step explanation:

4/625 x 625/9 = 4 x 1 / 5 x 5 x 5 x 1 = 4/625. The cross cancellation did not change the result.

Yes, any ramp with an angle of 4.8 degrees will have a slope ratio of 1:12.

The slope ratio is the ratio of the vertical rise to the horizontal run of the ramp, and it is equivalent to the tangent of the angle of inclination of the ramp.

The tangent of 4.8 degrees is approximately 0.0084, which means that for every 1 unit of vertical rise, there is 0.0084 units of horizontal run. To convert this to a ratio, we can multiply both sides by 100 to get:

1 unit of rise : 100 x 0.0084 = 0.84 units of run

Simplifying this ratio by dividing both sides by 0.84, we get:

1 unit of rise : 1.19 units of run

which is equivalent to a slope ratio of 1:12 (since 12 = 1/0.084). Therefore, any ramp with an angle of 4.8 degrees will have a slope ratio of 1:12.

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a. The function that would best model the population over time is Exponential decay

b. write an equation that models the changing population over time  P(t) = [tex]72,000 * e^(-0.035t)[/tex]

a. Exponential rot (Exponential decay) work would best demonstrate the populace over time.

Usually, the populace has diminished from 72,000 to 70,379 in fair 2 years, which could be a generally brief time period. Also, city authorities anticipate the populace to level off at 60,000, which is a sign of exponential rot.

b. The exponential rot work can be composed as:

P(t) = P0 *[tex]e^(-kt)[/tex]

Where P(t) is the populace at time t, P0 is the starting populace, e is the scientific steady around rise to 2.718, and k is the rot consistent.

Utilizing the given data, able to substitute the values:

P(0) = 72,000 (populace in 2018)

P(2) = 70,379 (populace in 2020)

To illuminate for k, able to utilize the equation:

k = ln(P0/P(t))/t

k = ln(72,000/70,379)/2

k ≈ 0.035

Subsequently, the condition that models the changing populace over time is:

P(t) = [tex]72,000 * e^(-0.035t)[/tex]

where t is the time in a long time since 2018. 

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The mean of the sampling distribution is 0.12 and the standard deviation is 0.033

(a) The mean of the sampling distribution of p^ is equal to the population proportion, which is p = 0.12. The standard deviation of the sampling distribution of p^ is given by the formula:

σ = sqrt[(p(1-p))/n]

where n is the sample size. Plugging in the values, we get:

σ = sqrt[(0.12)(0.88)/100] = 0.033

Therefore, the mean of the sampling distribution is 0.12 and the standard deviation is 0.033.

(b) The standard deviation from part (a) represents the amount of variability we expect to see in the sampling distribution of p^ due to chance.

It tells us how much we would expect p^ to vary from sample to sample, if we were to repeat the sampling process many times.

(c) Yes, it would be appropriate to use a normal distribution to model the sampling distribution of p^, because the sample size n is large enough (n=100) for the Central Limit Theorem to apply.

According to the Central Limit Theorem, the sampling distribution of p^ will be approximately normal with mean p and standard deviation σ/sqrt(n), as long as the sample size is sufficiently large.

In this case, the sample size is large enough, so we can use a normal distribution to model the sampling distribution of p^.

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Each equation should be matched to the situation it represents as follows;

"Leilah has not yet studied 600 of her 2400 flashcards. She studies 40 new cards each day." ⇒ 2400 - 40x = 600

"Stetson rents studio space for $600 a month for music lessons. He charges his students $40 per hour and earned a profit of $2400 this month." ⇒ 40x - 600 = 2400

"A kit contains 600 letter tiles and 40 number tiles. Each tile has the same mass, and the kit has a total mass of 2400 g." ⇒ (600 + 40)x = 2400.

In this scenario and exercise, the independent variable (domain or input value) would be represented by the variable x, and then each of the situations described by the word sentence (problem) would be translated into an algebraic equation or linear function as follows;

Since Leilah studies 40 new cards per day, but hasn't studied 600 of her 2400 flashcards yet, a linear function to model or represent this situation is given by;

2400 - 40x = 600

The rent for Stetson's studio space is $600 per month and he charges his students $40 each hour while earning a profit of $2400 this month, a linear function to model or represent this situation is given by;

40x - 600 = 2400

Since this kit with a total mass of 2400rams contains 600 letter tiles and 40 number tiles, and each of the tiles have the same mass, the required linear function is given by;

(600 + 40)x = 2400.

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Answer:

3595.5

Step-by-step explanation:

50% of the beads Nolan received were orange.

To find the percentage of beads that were orange, we need to divide the number of orange beads by the total number of beads and then multiply by 100.

Percentage of orange beads = (Number of orange beads / Total number of beads) * 100

In this case, Nolan received a total of 86 beads, and 43 of them were orange.

Percentage of orange beads = (43 / 86) * 100

Calculating this expression:

Percentage of orange beads = 0.5 * 100

Percentage of orange beads = 50%

Therefore, Out of total number of  beads Nolan received, 50% were orange.

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Part A: The coefficient of the 4th degree term in the Taylor polynomial for f(x) = sin(4x) centered at x = 0 is -1/3! = -1/6.

Part B: Using a 4th degree Taylor polynomial for sin(x) centered at x = 0, we can write g(x) = sin(0.8) ≈ P4(0.8), where P4(0.8) is the 4th degree Taylor polynomial for sin(x) evaluated at x = 0.8.

Evaluating P4(0.8) using the formula for the Taylor series coefficients of sin(x), we get P4(0.8) = 0.8 - 0.008 + 0.00004 - 0.0000014 ≈ 0.78333. This estimate is very close to 1 because sin(0.8) is close to 1, and the Taylor series for sin(x) converges very rapidly for values of x close to 0.

Part C: The series { (-1)n / (2n + 1) } has a partial sum S when x = 1. To find an interval |S - S5| = R5| for which the actual sum exists, we can use the alternating series test. The alternating series test states that if the terms of a series alternate in sign, decrease in absolute value, and approach zero, then the series converges.

Since the terms of the series { (-1)n / (2n + 1) } alternate in sign and decrease in absolute value, we know that the series converges. To find an interval |S - S5| = R5|, we can use the remainder formula for alternating series, which states that |Rn| ≤ a_n+1, where a_n+1 is the first neglected term in the series.

Since the terms of the series decrease in absolute value, we know that a_n+1 ≤ |a_n|. Therefore, we have |R5| ≤ |a6| = 1/7!, which means that the actual sum of the series exists in the interval S - 1/7! ≤ S5 ≤ S + 1/7!. Therefore, an interval for which the actual sum exists is [S - 1/7!, S + 1/7!].

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But there seems to be some missing information in your question. Please provide more context or details so that I can assist you accurately.
Hi! I'd be happy to help you evaluate the indefinite integral. Based on the provided terms and information, it seems like you want to evaluate the following integral:

∫x * sin(e * cos(x)) dx
To solve this integral, we can use integration by parts, which is defined as:
∫u dv = u * v - ∫v du
Let's choose u = x and dv = sin(e * cos(x)) dx. Then, we need to find du and v:
du = dx
v = ∫sin(e * cos(x)) dx

Unfortunately, the integral for v does not have a simple closed-form expression. However, you can use numerical methods or software (like Wolfram Alpha) to approximate it. Once you have an approximation for v, you can plug it back into the integration by parts formula to obtain an approximation of the original integral:
∫x * sin(e * cos(x)) dx ≈ x * v - ∫v dx

Keep in mind that this is an indefinite integral, so don't forget to add the constant of integration, C, to your final answer.

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After the first year, Carlotta will have $124.80 in the investment portion of her policy.

This is calculated by taking 40% of her annual premium

($780 x 0.40 = $312),

2.2% ($312 x 0.022 = $6.84).

So the total amount in the investment portion is

$312 + $6.84 = $318.84, rounded to the nearest cent, which is $124.80.

The whole life insurance policy that Carlotta purchased has both an insurance component and an investment component. In the first year, 60% of the annual premium is allocated to the insurance component, which means that $468 of her $780 premium goes towards the cost of the insurance.

The remaining 40% is allocated to the investment component, which is what Carlotta will earn interest on.

At a rate of 2.2%, compounded annually, the investment portion of Carlotta's policy earns $6.84 in interest after the first year. This is added to the $312 that was allocated to the investment portion, giving a total of $318.84.

This means that Carlotta has $124.80 in the investment portion of her policy after the first year. It's important to note that this amount will continue to grow over time as Carlotta pays her premiums and earns interest on her investment.

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Answer: £326

Step-by-step explanation:

Step 1: Calculate the cost per jumper

To find out how much Ryan spent on each jumper, we divide the total cost by the number of jumpers.

[tex]\frac{130}{40} = 3.25[/tex]

This gives us a cost of £3.25 per jumper.

Step 2: Calculate the revenue from selling 80% of the jumpers

Ryan sells 80% of the 40 jumpers, so:

[tex]\text{0.8 x 40 = 32}[/tex]

So he sold 32 Jumpers.

He sells each jumper for £12:

[tex]\text{32 x 12 = 384}[/tex]

So his revenue from selling these jumpers is £384

Step 3: Calculate the revenue from selling the remaining jumpers on the Buy one get one half price offer

Ryan has 8 jumpers left after selling 80% of them. He puts these on a Buy one get one half price offer, which means that for every jumper sold at full price, he sells another one at half price.

This means that he sells 4 jumpers at full price (£12 each) and 4 jumpers at half price (£6 each).

His revenue from selling these jumpers is:

[tex]\text{(4 x 12) + (4 x 6) = 72}[/tex]

Step 4: Calculate the total revenue

Ryan's total revenue is the sum of the revenue from selling 80% of the jumpers and the revenue from selling the remaining jumpers on the Buy one get one half price offer.

This is:

[tex]\text{384 + 72 = 456}[/tex]

So Ryan's total revenue is £456

Step 5: Calculate the total cost

Ryan's total cost is the amount he spent on buying the jumpers, which is £130.

Step 6: Calculate the profit

Ryan's profit is the difference between his total revenue and his total cost:

[tex]\text{456 - 130 = 326}[/tex]

Therefore, Ryan makes a profit of £326.

a) Para calcular este límite, podemos dividir tanto el numerador como el denominador por n² y luego aplicar la regla de L'Hôpital:

lim n → ∞ [(6 - 4n²)/(2n²)]

= lim n → ∞ [6/(2n²) - (4n²)/(2n²)]

= lim n → ∞ [3/n² - 2]

= -2

Por lo tanto, el límite es -2.

b) Podemos dividir tanto el numerador como el denominador por n³ para simplificar el límite:

lim n → ∞ [(4n² + 3n - 2)/(2n³ - 4n)]

= lim n → ∞ [(4/n + 3/n² - 2/n³)/(2/n² - 4/n²)]

= lim n → ∞ [(4 + 3/n - 2/n²)/(2 - 4/n)]

= lim n → ∞ [(4n + 3 - 2n²)/(2n² - 4)]

= lim n → ∞ [-2n²/(2n² - 4)]

= -1

Por lo tanto, el límite es -1.

c) Podemos dividir tanto el numerador como el denominador por n³ para simplificar el límite:

lim n → ∞ [(2n³ - 4n)/(4n)]

= lim n → ∞ [(2n² - 4)/(4)]

= lim n → ∞ [(n² - 2)/2]

= +∞

Por lo tanto, el límite es +∞.

d) Podemos dividir tanto el numerador como el denominador por x⁴ para simplificar el límite:

lim x → ∞ [-8x⁴ + 2]/[2x² + 4]

= lim x → ∞ [-8 + 2/x⁴]/[2/x² + 4/x⁴]

= -4/1

= -4

Por lo tanto, el límite es -4.

The perimeter of the shape is 53.7 units

Perimeter is a math concept that measures the total length around the outside of a shape.

A theorem of circle geometry states that the tangent from a point on a circle are equal.

Therefore the base sides is calculated as

9.9 + 3.2

= 13.1

since the perimeter is the addition of all the sides then;

P = 13.1 + 21.9 + 18.7

P = 53.7

therefore the perimeter of the triangle is 53.7

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The quadrilateral formed by the vertices A(-2, 4), B(2, 1), C(-1, -3), and D(-5, 0) has an area of 21/2 square units.

To find the area of the quadrilateral with the given coordinates A(-2, 4), B(2, 1), C(-1, -3), D(-5, 0), we can use the formula for the area of a quadrilateral in the coordinate plane:

Area = |(1/2)(x1y2 + x2y3 + x3y4 + x4y1 - x2y1 - x3y2 - x4y3 - x1y4)|

where (x1, y1), (x2, y2), (x3, y3), and (x4, y4) are the coordinates of the vertices of the quadrilateral.

Substituting the given coordinates, we get:

Therefore, the area of the quadrilateral with the given coordinates is 21/2 square units.

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The probability of drawing a star-shaped chip is 5/8.

The theoretical probability of drawing a star-shaped circular chip from the bag is 5/8 or 0.625 or 62.5%. Out of the total of eight circular chips, five are stars, and three are squares.

Therefore, the probability of drawing a star-shaped chip is the ratio of the number of star-shaped chips to the total number of chips in the bag, which is 5/8.

To understand this conceptually, we can think of probability as a fraction where the numerator is the number of favorable outcomes (in this case, drawing a star-shaped chip) and the denominator is the total number of possible outcomes (all the circular chips in the bag).

Thus, the theoretical probability of drawing a star-shaped chip is 5/8 because there are five star-shaped chips out of the total eight circular chips in the bag.

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The amount of total and per-unit overhead allocated to each of the three products using overhead rate is equal to,

Total Per Unit Factory Overhead ,  Cost Factory Overhead Cost

Flutes $530   $1,060,000

Clarinets $795    $1,192,500

Oboes $397.5     $695,625

Total $1,722.5    $2,948,125

Budgeted factory overhead cost = $2,948,125

The single plantwide overhead rate

= Dividing the budgeted factory overhead cost by the total budgeted direct labor hours.

For this,

Flutes= 2,000×2

         = 4,000 hours

Clarinets= 1,500×3

              = 4,500 hours

Oboes= 1,750×1.5

          = 2,625 hours

Total direct labor hours = 11,125

Substitute the value we have,

⇒ Single plantwide overhead rate = $2,948,125 / (2,000 x 2.0 + 1,500 x 3.0 + 1,750 x 1.5)

= $2,948,125 / 11,125

= $265 per direct labor hour

To allocate overhead to each product

=Multiply the overhead rate by the budgeted direct labor hours per unit for each product.

Substitute the value we have,

Flutes,

$265 x 2.0 = $530 total overhead cost, $265 per unit

Clarinets,

$265 x 3.0 = $795 total overhead cost, $265 per unit

Oboes,

$265 x 1.5 = $397.5 total overhead cost, $265 per unit

And

Total factory overhead cost allocated = Estimated manufacturing overhead rate×  Actual amount of allocation base

For,

Flutes

= 4,000× 265

= $1,060,000

Clarinets

= 4,500×265

= $1,192,500

Oboes

= 2,625×265

= $695,625

This implies,

The total and per-unit overhead allocated to each product, rounded to the nearest dollar is,

Total Per Unit Factory Overhead ,  Cost Factory Overhead Cost

Flutes $530 $1,060,000

Clarinets $795 $1,192,500

Oboes $397.5  $695,625

Total $1,722.5 $2,948,125

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The above question is incomplete, the complete question is:

Bach Instruments Inc. makes three musical instruments: flutes, clarinets, and oboes. The budgeted factory overhead cost is $2,948,125. Overhead is allocated to the three products on the basis of direct labor hours. The products have the following budgeted production volume and direct labor hours per unit:

Budgeted Production Volume Direct Labor Hours Per Unit

Flutes 2,000 units 2.0

Clarinets 1,500 3.0

Oboes 1,750 1.5

a. Determine the single plantwide overhead rate.

$ per direct labor hour

b. Use the overhead rate in (a) to determine the amount of total and per-unit overhead allocated to each of the three products, rounded to the nearest dollar.

Total Per Unit

Factory Overhead Cost Factory Overhead Cost

Flutes $ $

Clarinets

Oboes

Total $

Check the picture below.

The maximum rate of change occurs in the direction of this unit vector.

(a) To find the maximum rate of change of f at point P(1,0), we need to find the gradient of f at that point and then find its magnitude. The direction of maximum increase is given by the unit vector in the direction of the gradient.

The gradient of f is:

∇f(x,y) = <y cos(xy), x cos(xy)>

At point P(1,0), we have:

∇f(1,0) = <0, cos(0)> = <0, 1>

The magnitude of the gradient is:

||∇f(1,0)|| = sqrt([tex]0^2[/tex] +[tex]1^2[/tex]) = 1

Therefore, the maximum rate of change of f at point P is 1, and it occurs in the direction of the unit vector in the direction of the gradient:

u = <0, 1>/1 = <0, 1>

So the maximum rate of change occurs in the y-direction.

(b) To find the maximum rate of change of f at point P(8,1.3), we need to find the gradient of f at that point and then find its magnitude. The direction of maximum increase is given by the unit vector in the direction of the gradient.

The gradient of f is:

∇f(x,y,z) = <2x, 2y, 2z>

At point P(8,1.3), we have:

∇f(8,1.3) = <16, 2.6, 2(1.3)> = <16, 2.6, 2.6>

The magnitude of the gradient is:

||∇f(8,1.3)|| = sqrt[tex](16^2 + 2.6^2 + 2.6^2)[/tex]= sqrt(275.56) ≈ 16.6

Therefore, the maximum rate of change of f at point P is approximately 16.6, and it occurs in the direction of the unit vector in the direction of the gradient:

u = <16, 2.6, 2.6>/16.6 ≈ <0.963, 0.157, 0.157>

So the maximum rate of change occurs in the direction of this unit vector.

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Answer:

The order from smallest to largest price per package is: December, April, February.

Step-by-step explanation:

For the sale in February:

- Price per package = $4.49 ÷ 3 = $1.50 per package

For the sale in April:

- Price per package = $7.39 ÷ 5 = $1.48 per package

For the sale in December:

- Price per package = $5.88 ÷ 4 = $1.47 per package